Complete monotonicity conjecture for even powers of Baskakov functions

Let $r>1$ be an even integer and let $\alpha>0$. Define

$$f_{\alpha}^{[r]}(x)=(1+x)^{-r\alpha}\sum_{k=0}^{\infty}\binom{-\alpha}{k}^{r}\left(\frac{x}{1+x}\right)^{rk},\qquad x\geq 0.$$

For positive $c$ and $n\in\mathbb{N}$, also define

$$p_{n,k}^{[c]}(x)=\binom{-n/c}{k}\left(-cx\right)^k\left(1+cx\right)^{-n/c-k}$$

and

$$\psi_{n,c}^{[r]}(x)=\sum_{k=0}^{\infty}\left(p_{n,k}^{[c]}(x)\right)^r,\qquad x\geq 0.$$

Even-power complete-monotonicity conjecture. The function $f_{\alpha}^{[r]}$ is completely monotonic on $[0,\infty)$. In particular, $\psi_{n,c}^{[r]}$ is completely monotonic on $[0,\infty)$. The paper presents this as a conjecture after noting that numerical simulations suggest the property for even $r$, while its results establish the case $r=2$; the general even-power case remains open in the supplied text.