Conjectured vanishing sum identity for central factorial numbers of the first kind

Prove that for all integers n ≥ 1 and m ∈ {0,1,…,n−1}, the weighted finite sum over central factorial numbers of the first kind t(2n, 2(n−k)) with binomial-type coefficients equals zero, as stated in equation (4.41). In particular, the conjecture asserts that a specific combination of the terms t(2n, 2(n−k)) for k = 0,…,m, multiplied by coefficients depending on n, m, and k (including factors of powers of 4 and binomial coefficients in 2(n−k)−1 and 2(n−m−1)), sums to 0.

Background

This conjecture is presented as an immediate consequence of a preceding identity (equation (4.40)) that arises from comparing two different expressions for certain integrals and sums involving beta- and zeta-values. The structure suggests a hidden orthogonality or cancellation among central factorial numbers of the first kind when combined with specific combinatorial weights.

While the paper derives supporting identities that motivate this claim, a direct proof is not provided, and the author explicitly presents it as a conjecture.