Kronecker-delta sum conjecture for the generalized alternating hyperharmonic coefficients

Let $r,m\in\mathbb N$, let $f(r,s_{1},s_{2})$ be the index bound occurring in the coefficient recursion, and let $\delta_{mn}$ denote the Kronecker delta, with $\delta_{mm}=1$ and $\delta_{mn}=0$ for $m\neq n$. The coefficient sum conjecture. For $0\leq m\leq f(r,s_{1},s_{2})$, we conjecture that

$$\sum_{j=0}^{m}\sum_{k=0}^{3}b(r,s_{1},s_{2},m-j,j,k)=\delta_{m0}.$$

This proposed identity is intended to capture another structural property of the coefficients appearing in the reduction of Euler sums of generalized alternating hyperharmonic numbers; no resolution is supplied.