By modifying the approach in my answer here, I found thousands of solutions, including:
dormy fling jaups vetch
You can also make "fetch" happen:
fetch groks laund wimpy
And $15$ with two particularly relevant words fixed:
cheat flump vying words
cheat flimp gunky words
cheat flung jimpy words
cheat fling jumpy words
cheat flunk pigmy words
bumpy cheat fling words
blunk cheat pigmy words
cheat jumby pling words
blunk cheat gimpy words
cheat flunk gimpy words
cheat plumb vying words
bingy cheat flump words
cheat fling jumby words
bungy cheat flimp words
blimp cheat gunky words
Explicitly, the modifications are:
- Restrict word.list to five-letter words that do not contain q, x, or z.
- Change the $3$ to $4$ in the equality constraint.
- Fix $y_\ell=1$ for the $15$ required letters.