Integrability conjecture for the Fourier transform of extremal functions

Let $\varphi_p$ be the extremal function for the Paley–Wiener extremal problem, and let $\widehat{\varphi}_p$ denote its Fourier transform. Fourier-transform regularity conjecture. If $1<p<\infty$, then $\widehat{\varphi}_p$ is a continuous integrable function on $(-\pi,\pi)$. The source explains that the analogous properties are automatic for $0<p\leq1$ and conjectures them in the strictly convex regime.