Shapiro's real-zero bound for Jensen polynomials

Let $p(x)$ be a degree-$k$ polynomial with real coefficients, and let $P_i(x)$ denote the Jensen polynomials defined in the surrounding discussion by the expansion of $|p(x+iy)|^2$ in even powers of $y$.

Jensen-polynomial zero-bound conjecture. For every such $p$ and every relevant index $i$,

$$\sharp_r(P_i(x))\leq\min\{\deg P_i(x),k\}.$$

The claim is posed after related experimental questions about the families $P_i$ and $G_i$; the source gives no resolution.