Clark–Ismail positivity conjecture for the functions $f_j$ and $g_j$

Let $j>1$ and $v>0$. Define

$$f_j(v)=\frac{\operatorname{d}^j}{\operatorname{d}v^j}\left(\frac{v^j}{1-\operatorname{e}^{-v}}\right),$$

and

$$g_j(v)=\left(\frac{1-\operatorname{e}^{-v}}{v}\right)^{j+1}\operatorname{e}^{jv}f_j(v).$$

Clark–Ismail's positivity conjecture. The functions $f_j(v)$, or equivalently $g_j(v)$, are positive on $(0,\infty)$ for all $j>1$. This conjecture concerns the positivity needed to establish complete monotonicity of the associated derivatives of polygamma functions; it was posed by Clark and Ismail after verifying the claim computationally for $2\le j\le16$, and its general status is not specified in the source.