An Explicit Conjectured Determinant Evaluation Whose Proof Would Make Me Happy (and the OEIS richer)

I conjecture a certain explicit determinant evaluation, whose proof would imply the solution of certain enumeration problem that I have been working on, and that I find interesting. I am pledging $500 to the OEIS Foundation (in honor of the prover!) for a proof, and $50 (in honor of the disprover)