The complex Gauss–Lucas-type conjecture for derivative convolutions

Let $p\in\mathbb{N}^*$ and let $P,Q\in\mathbb{C}[X]$ have degree $p$. Derivative-convolution Gauss–Lucas conjecture. The roots of

$$\sum_{k=0}^p P^{(k)}Q^{(p-k)}$$

lie in the convex hull of the roots of $P$ and $Q$. This is proposed as a natural complex analogue of the derivative-convolution real-rootedness conjecture and would imply that conjecture. No proof or disproof is given in the paper.