>> Please remove (= n 1) and replace (setq n 0) with (setq n (1- n)) to 
>> support the prefix arg with ‘M-N M-n’.
>
> When n > 0.
>

You mean this, right?

diff --git a/lisp/simple.el b/lisp/simple.el
index 63479e9ce0..0221881641 100644
--- a/lisp/simple.el
+++ b/lisp/simple.el
@@ -9698,6 +9698,14 @@ next-completion
    (let ((tabcommand (member (this-command-keys) '("\t" [backtab])))
          pos)
      (catch 'bound
+      (when (and (bobp)
+                 (> n 0)
+                 (get-text-property (point) 'mouse-face)
+                 (not (get-text-property (point) 'first-completion)))
+        (let ((inhibit-read-only t))
+          (add-text-properties (point) (1+ (point)) '(first-completion t)))
+        (setq n (1- n)))
+
        (while (> n 0)
          (setq pos (point))
          ;; If in a completion, move to the end of it.

It took me a few minutes to understand what you meant by "support the 
prefix arg" in this case.  And indeed with the previous version of the 
patch, M-x M-2 M-<down> selects the third completion candidate (2C-split), 
whereas with this patch the second one (2C-command) is selected.