Shape conjecture for Fourier transforms of extremal functions

Let $\varphi$ be a solution of the Paley–Wiener extremal problem, and let $\widehat{\varphi}$ denote its Fourier transform. Fourier-transform shape conjecture. The function $\widehat{\varphi}$ is nonnegative. Moreover,

(a) if $0<p<2$, then $\widehat{\varphi}$ is decreasing on $(0,\pi)$ and $\widehat{\varphi}(\xi)\to0$ as $\xi\to\pi^-$;

(b) if $2<p<\infty$, then $\widehat{\varphi}$ is increasing on $(0,\pi)$ and $\widehat{\varphi}(\xi)\to\infty$ as $\xi\to\pi^-$. The source motivates this from the resemblance of the extremal functions to explicit model functions.