Symmetry conjecture for the generalized alternating hyperharmonic coefficients

Let $r,m,j\in\mathbb N$, and let $f(r,s_{1},s_{2})$ be the index bound occurring in the coefficient recursion. For $0\leq m\leq f(r,s_{1},s_{2})$ and $0\leq j\leq f(r,s_{1},s_{2})-m$, let $b(r,s_{1},s_{2},m,j,k)$ denote the corresponding coefficient for $k\in\\{0,1,2,3\\}$. The coefficient symmetry conjecture. We conjecture that

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

This is one of three proposed structural properties of the coefficients that express Euler sums of generalized alternating hyperharmonic numbers in terms of classical alternating Euler sums; its resolution is not given in the supplied text.