Thank you, this is really nice!

@RichardStanley I have corrected the sum --- the power of $\beta$ should have been $j$ rather than $n$.

Yes, my sum has a falling rather than a rising power. But as observed in a comment in the earlier question, when the argument of the Pochhammer symbol is constant (not depending on n), there's not any difference in difficulty between falling and rising power.

There is a very nice recent survey of progress on these kinds of problems, by Griggs and Li, at link.springer.com/chapter/10.1007%2F978-3-319-24298-9_14. Thm 1.2 covers your updated question.