diotalevi provided the explanation of the problem. The following is a tested solution: 
...

our $rxNest;
$rxNest = qr{(?x)
   (
      \(
      [^()]*
      (?:
         (??{$rxNest})
         [^()]*
      )*
      \)
   )
   (?{ [ @{$^R}, $1 ] })
};

our $rxOnlyNested = qr{(?x)
   (?{ [] })
   ^
   [^()]*
   (?:
      $rxNest
      [^()]*
   )*
   \z
   (?{ @memoList = @{$^R} })
};

...

    if ($string =~ $rxOnlyNested)

...

[download]

Note that @memoList is now correct whether there is backtracking or not, thanks to the use of $^R.

Note that I used $1 instead of $+. Using $+ will negatively impact on the performance of all regexps without captures executed in that perl interpreter instance, whereever they are located.

Update: Removed unnecessary parens. (Copy and paste bug.)

In reply to Re: Recursive regular expression weirdness by ikegami
in thread Recursive regular expression weirdness by johngg

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