Monotonicity conjecture for the point-evaluation constants

Let $\mathscr{C}_p$ denote the point-evaluation constant associated with the Paley–Wiener space $PW^p$. Monotonicity conjecture. The function

$$p \mapsto \frac{\mathscr{C}_p}{p}$$

is strictly decreasing on $(0,\infty)$. If true, this hypothesis would likely yield a simpler proof of the relevant point-evaluation theorem and imply further bounds for $\mathscr{C}_p$.