Fischer–Kubitzke conjecture on eventual real-rootedness of Hadamard powers

Let $p$ be a polynomial with nonnegative coefficients, and let $\mathscr{W}(p)$ denote the polynomial obtained by the operator $\mathscr{W}$. Fischer–Kubitzke conjecture. For all sufficiently large positive integers $n$, the polynomial $\mathscr{W}(p^n)$ is real-rooted.

This conjecture concerns the behavior of the operator $\mathscr{W}$ under Hadamard powers and predicts eventual real-rootedness. It is presented as a reformulation of a conjecture posed by Fischer and Kubitzke; no resolution is supplied here.