The derivative-convolution real-rootedness conjecture

Let $p\in\mathbb{N}^*$ and let $P,Q\in\mathbb{R}[X]$ have degree $p$. Derivative-convolution conjecture. If $P$ and $Q$ are real-rooted, then

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

is real-rooted as well. This is presented as an equivalent formulation of the semi-symmetric hyperbolicity conjecture, so it is connected to the concavity problem through the real-rootedness of associated polynomials. It remains open in the paper.