Uniqueness-set conjecture for the extremal zero set

Let $\varphi_p$ be the unique solution of the extremal problem, let $\mathscr{Z}(\varphi_p)$ denote its zero set, and let $PW^p$ be the Paley–Wiener space. A set is a uniqueness set for $PW^p$ if the only function in $PW^p$ vanishing on it is the zero function. Uniqueness-set conjecture.

$$\mathscr{Z}(\varphi_p)\cup\{0\}\text{ is a uniqueness set for }PW^p\quad\Longleftrightarrow\quad p\geq1.$$

The assertion has been verified in the source for $p\geq2$ and remains to be checked below $2$.