co.combinatorics – Conjecture on bernoulli numbers and binomial coefficients

I posted a question on mathstackexchange here regarding a conjecture relating sums of bernoulli numbers and binomial coefficients

https://math.stackexchange.com/questions/4116414/conjecture-on-bernoulli-numbers-and-binomial-coefficients

I would greatly appreciate any help. I can go through the work of retyping the question here, if it’s easier, I’m not sure what the etiquette is.

Thanks!