Sign conjecture for the coefficients of generalized hyperharmonic numbers

Let $a(r,m,\ell)$ be the coefficients appearing in the expansion under consideration, and let $\operatorname{sgn}(x)$ be the signum function:

$$\operatorname{sgn}(x)=\begin{cases}1&x>0,\\0&x=0,\\-1&x<0.\end{cases}$$

For $r,m\in\mathbb N$ with $0\leq m\leq r-1$, Sign conjecture.

$$\operatorname{sgn}(a(r,m,\ell))=(-1)^m\quad (0\leq\ell\leq r-1-m).$$

In particular,

$$a(r,m,\ell)\ne 0\quad (0\leq\ell\leq r-1-m).$$

This conjecture predicts a fixed sign in each coefficient row and nonvanishing; the source gives no resolution.