These first four defcustoms behave as I expect.

The first has a default value of nil - empty alist.  The second has a
default value of (("key1" :x ("a" "b") ("q"))).  The third had a default
value of (("key1" :x () :w ("q"))) - one of the plist-entry values is
empty.  The fourth has a default value of (("key1")) - empty plist.

The option should be an alist with elements that have a string key and a
plist value.  Each plist element has a keyword key and a list of string
values.  The alist can be empty, any plist can be empty, and any
plist-entry value can be empty.

;; OK
(defcustom myvar ()
  "..."
  :group 'emacs
  :type '(alist :key-type (string :tag "Alist key (string):")
	        :value-type
		(plist :key-type symbol
                       :options (:x :y :z)
                       :value-type (repeat string))))

;; OK
(defcustom myvar '(("key1" :x ("a" "b") :w ("q")))
  "..."
  :group 'emacs
  :type '(alist :key-type (string :tag "Alist key (string):")
	        :value-type
		(plist :key-type
                       (restricted-sexp :match-alternatives (keywordp)
                                        :tag "Plist key (keyword)")
                       :options (:x :y :z)
                       :value-type (repeat string))))

;; OK
(defcustom myvar '(("key1" :x () :w ("q")))
  "..."
  :group 'emacs
  :type '(alist :key-type (string :tag "Alist key (string):")
	        :value-type
		(plist :key-type
                       (restricted-sexp :match-alternatives (keywordp)
                                        :tag "Plist key (keyword)")
                       :options (:x :y :z)
                       :value-type (repeat string))))

;; OK
(defcustom myvar '(("key1"))
  "..."
  :group 'emacs
  :type '(alist :key-type (string :tag "Alist key (string):")
	        :value-type
		(plist :key-type
                       (restricted-sexp :match-alternatives (keywordp)
                                        :tag "Plist key (keyword)")
                       :options (:x :y :z)
                       :value-type (repeat string))))

The first OK example specifies the type of the plist key as just symbol,
not as keywordp.  But I want to constrain it to be a symbol that's a
keyword.  The problem occurs when I try to do that AND I use a default
value of nil (see next).

This one is no good - that's the bug:

;; When click initial INS get prompted for a Lisp expression, get warnings,
;; and Customize buffer contents are changed to just INS DEL while prompting.
;;
(defcustom myvar ()
  "..."
  :group 'emacs
  :type '(alist :key-type (string :tag "Alist key (string):")
	        :value-type
		(plist :key-type
                       (restricted-sexp :match-alternatives (keywordp)
                                        :tag "Plist key (keyword)")
                       :options (:x :y :z)
                       :value-type (repeat string))))

In this case the default value is nil, but the defcustom also specifies
the type of plist values as keywordp.  I think this definition should
work fine.  But when you click the sole INS button at the outset you're
prompted for a Lisp expression and you simultaneously get a warning that
a plist key is not of the right type.  If you enter a Lisp expression
(e.g. just 42 or whatever) then the Customize buffer appears as it
should.  I think (1) you shouldn't be prompted for anything and (2) you
shouln't get any warning of anything.

  Warning (widget-bad-default-value): 
  A widget of type restricted-sexp has a bad default value.
  value: nil
  match function: widget-restricted-sexp-match
  match-alternatives: (keywordp) Disable showing Disable logging

See attached screenshot, which shows the prompt as well as the
Customize buffer during prompting and the *Warnings* buffer.

In GNU Emacs 28.2 (build 2, x86_64-w64-mingw32)
 of 2022-09-13 built on AVALON
Windowing system distributor 'Microsoft Corp.', version 10.0.19044
System Description: Microsoft Windows 10 Pro (v10.0.2009.19044.2251)

Configured using:
 'configure --with-modules --without-dbus --with-native-compilation
 --without-compress-install CFLAGS=-O2'

Configured features:
ACL GIF GMP GNUTLS HARFBUZZ JPEG JSON LCMS2 LIBXML2 MODULES NATIVE_COMP
NOTIFY W32NOTIFY PDUMPER PNG RSVG SOUND THREADS TIFF TOOLKIT_SCROLL_BARS
XPM ZLIB

(NATIVE_COMP present but libgccjit not available)