Boundary-sum conjecture for the coefficients of generalized hyperharmonic numbers

Let $a(r,m,\ell)$ be the coefficients appearing in the expansion under consideration. Boundary-sum conjecture. For $r\in\mathbb N$,

$$\sum_{\ell=0}^{r-1}a(r,0,\ell)=r\quad (r\geq 1),$$

and

$$\sum_{\ell=0}^{r-1}a(r,\ell,0)=0\quad (r\geq 2).$$

This is one of four conjectures suggested by the tabulated values for small $r$; the source does not state whether it has been resolved.