Uniqueness conjecture for the Paley–Wiener extremal problem below $p=1$

For $0<p<\infty$, consider the extremal problem defining the Paley–Wiener extremal function, and call a function attaining the extremum a solution. Uniqueness conjecture. The extremal problem has a unique solution for $0<p<1$. The source describes this as the most basic unresolved problem about the extremal functions.