Korevaar-type conjecture for imaginary-axis point evaluation

Fix $0<p<\infty$, and let $\mathscr{K}_p$ be the best constant in Korevaar's inequality for point evaluation on the imaginary axis, namely $|f(iy)|^p\leq \mathscr{K}_p\operatorname{sinc}(i\pi p y)\|f\|_p^p$ for all $y\geq0$ and $f\in PW^p$. Imaginary-axis point-evaluation conjecture.

$$\mathscr{K}_p=\begin{cases}\mathscr{C}_p,&0<p<2,\\ p/2,&2\leq p<\infty.\end{cases}$$

This adjusts Korevaar's original conjecture, which the source notes is refuted.