I forgot to mention: the driver in use is ATI, not Radeon.

> I don't know for sure, but that Radeon RX 580 card could be the
> problem. I've set up a machine with Debian 9 and a Radeon R9 285 in the
> past, and it was a pain to make it work -- it would not work at all
> without the firmware blobs, and even then, I had to use a custom kernel
> from AMD to get it working correctly...

In my case the RX580 work very well Linux libre 4.20.4.

Maybe you Debian 9 was too old (kernel + mesa).

I'm still confused about the state of FOSS-ness with AMD.  Is any blob required
to run them at all?  Are is it only specific to AMDGPU-PRO?

> If you have a 2nd (old nVidia would be the best) card laying around, I'd
> suggest trying that -- if it works, you'll know the problem is with the
> amggpu driver.

Well, everything seems to work, I can control it my computer over SSH, so it's
most probably the RX 580 to blame.

I have also tried connecting my screen both over HDMI and DisplayPort, should that
make a difference.  It doesn't.

> Would it be possible for you to SSH into it to read the kernel messages during wakeup?

I did, and here is the relevant part of the dmesg output:

--8<---------------cut here---------------start------------->8---
[ 2976.265296] PM: suspend entry (deep)
[ 2976.265299] PM: Syncing filesystems ... done.
[ 2976.624270] Freezing user space processes ... (elapsed 0.001 seconds) done.
[ 2976.625607] OOM killer disabled.
[ 2976.625656] Freezing remaining freezable tasks ... (elapsed 0.001 seconds) done.
[ 2976.626804] printk: Suspending console(s) (use no_console_suspend to debug)
[ 2976.627367] serial 00:04: disabled
[ 2976.627803] r8169 0000:07:00.0 enp7s0: Link is Down
[ 2976.657161] sd 5:0:0:0: [sdb] Synchronizing SCSI cache
[ 2976.657192] sd 4:0:0:0: [sda] Synchronizing SCSI cache
[ 2976.657629] sd 4:0:0:0: [sda] Stopping disk
[ 2976.657724] sd 5:0:0:0: [sdb] Stopping disk
[ 2977.617490] ACPI: Preparing to enter system sleep state S3
[ 2977.941844] PM: Saving platform NVS memory
[ 2977.941882] Disabling non-boot CPUs ...
[ 2977.958842] smpboot: CPU 1 is now offline
[ 2977.974825] smpboot: CPU 2 is now offline
[ 2977.990581] smpboot: CPU 3 is now offline
[ 2978.014817] smpboot: CPU 4 is now offline
[ 2978.026540] smpboot: CPU 5 is now offline
[ 2978.042508] smpboot: CPU 6 is now offline
[ 2978.058498] smpboot: CPU 7 is now offline
[ 2978.074474] smpboot: CPU 8 is now offline
[ 2978.090460] smpboot: CPU 9 is now offline
[ 2978.106384] smpboot: CPU 10 is now offline
[ 2978.122308] smpboot: CPU 11 is now offline
[ 2978.122935] ACPI: Low-level resume complete
[ 2978.122976] PM: Restoring platform NVS memory
[ 2978.123526] Enabling non-boot CPUs ...
[ 2978.123607] x86: Booting SMP configuration:
[ 2978.123608] smpboot: Booting Node 0 Processor 1 APIC 0x1
[ 2978.125878]  cache: parent cpu1 should not be sleeping
[ 2978.125952] microcode: CPU1: patch_level=0x08008206
[ 2978.126084] CPU1 is up
[ 2978.126110] smpboot: Booting Node 0 Processor 2 APIC 0x2
[ 2978.128338]  cache: parent cpu2 should not be sleeping
[ 2978.128411] microcode: CPU2: patch_level=0x08008206
[ 2978.128491] CPU2 is up
[ 2978.128500] smpboot: Booting Node 0 Processor 3 APIC 0x3
[ 2978.130727]  cache: parent cpu3 should not be sleeping
[ 2978.130792] microcode: CPU3: patch_level=0x08008206
[ 2978.130880] CPU3 is up
[ 2978.130889] smpboot: Booting Node 0 Processor 4 APIC 0x4
[ 2978.133112]  cache: parent cpu4 should not be sleeping
[ 2978.133179] microcode: CPU4: patch_level=0x08008206
[ 2978.133269] CPU4 is up
[ 2978.133279] smpboot: Booting Node 0 Processor 5 APIC 0x5
[ 2978.135516]  cache: parent cpu5 should not be sleeping
[ 2978.135579] microcode: CPU5: patch_level=0x08008206
[ 2978.135678] CPU5 is up
[ 2978.135687] smpboot: Booting Node 0 Processor 6 APIC 0x8
[ 2978.137993]  cache: parent cpu6 should not be sleeping
[ 2978.138064] microcode: CPU6: patch_level=0x08008206
[ 2978.138219] CPU6 is up
[ 2978.138230] smpboot: Booting Node 0 Processor 7 APIC 0x9
[ 2978.140518]  cache: parent cpu7 should not be sleeping
[ 2978.140589] microcode: CPU7: patch_level=0x08008206
[ 2978.140734] CPU7 is up
[ 2978.140744] smpboot: Booting Node 0 Processor 8 APIC 0xa
[ 2978.143023]  cache: parent cpu8 should not be sleeping
[ 2978.143094] microcode: CPU8: patch_level=0x08008206
[ 2978.143251] CPU8 is up
[ 2978.143262] smpboot: Booting Node 0 Processor 9 APIC 0xb
[ 2978.145540]  cache: parent cpu9 should not be sleeping
[ 2978.145610] microcode: CPU9: patch_level=0x08008206
[ 2978.145769] CPU9 is up
[ 2978.145779] smpboot: Booting Node 0 Processor 10 APIC 0xc
[ 2978.148060]  cache: parent cpu10 should not be sleeping
[ 2978.148132] microcode: CPU10: patch_level=0x08008206
[ 2978.148312] CPU10 is up
[ 2978.148324] smpboot: Booting Node 0 Processor 11 APIC 0xd
[ 2978.150602]  cache: parent cpu11 should not be sleeping
[ 2978.150673] microcode: CPU11: patch_level=0x08008206
[ 2978.150872] CPU11 is up
[ 2978.152067] ACPI: Waking up from system sleep state S3
[ 2978.230420] usb usb1: root hub lost power or was reset
[ 2978.230422] usb usb2: root hub lost power or was reset
[ 2978.230633] sd 4:0:0:0: [sda] Starting disk
[ 2978.230636] sd 5:0:0:0: [sdb] Starting disk
[ 2978.231175] serial 00:04: activated
[ 2978.544675] ata2: SATA link down (SStatus 0 SControl 330)
[ 2978.545981] ata9: SATA link down (SStatus 0 SControl 300)
[ 2978.545984] ata1: SATA link down (SStatus 0 SControl 330)
[ 2978.638465] usb 1-9: reset full-speed USB device number 3 using xhci_hcd
[ 2978.705962] ata6: SATA link up 6.0 Gbps (SStatus 133 SControl 300)
[ 2978.706736] ata6.00: configured for UDMA/133
[ 2978.895739] r8169 0000:07:00.0 enp7s0: Link is Down
[ 2979.134479] usb 1-8: reset low-speed USB device number 2 using xhci_hcd
[ 2979.555278] OOM killer enabled.
[ 2979.555329] Restarting tasks ... done.
[ 2979.557153] PM: suspend exit
[ 2979.773654] IPv6: ADDRCONF(NETDEV_UP): enp7s0: link is not ready
[ 2979.773926] Generic PHY r8169-700:00: attached PHY driver [Generic PHY] (mii_bus:phy_addr=r8169-700:00, irq=IGNORE)
[ 2979.876671] IPv6: ADDRCONF(NETDEV_UP): enp7s0: link is not ready
[ 2981.488341] r8169 0000:07:00.0 enp7s0: Link is Up - 100Mbps/Full - flow control rx/tx
[ 2981.488353] IPv6: ADDRCONF(NETDEV_CHANGE): enp7s0: link becomes ready
[ 2984.946005] ata5: SATA link up 6.0 Gbps (SStatus 133 SControl 300)
[ 2984.947770] ata5.00: configured for UDMA/100
--8<---------------cut here---------------end--------------->8---

No error, but no trace of ATI/RADEON either.

-- 
Pierre Neidhardt
https://ambrevar.xyz/