Factorial decay conjecture for roots of sparse polynomials

Let $\mathcal{F}_1(p,t)$ be the family of polynomials over $\mathbb{F}_p$ with exactly $t$ nonzero terms and normalized as in the paper. Let $\mathcal{F}_1(p,t,r)$ denote the subfamily whose members have exactly $r$ distinct nonzero roots, and write $R(f)$ for the number of such roots.

Factorial decay conjecture. There exists a constant $\gamma>0$ such that

$$\frac{|\mathcal{F}_1(p,t,r)|}{|\mathcal{F}_1(p,t)|}\leq\left(\frac{1}{r!}\right)^\gamma$$

for all prime $p$, $t\in\mathbb{N}$, and $r\in\mathbb{N}$.

This conjecture formalizes the heuristic that the number of nonzero terms and the number of nonzero roots are statistically independent for a random sparse polynomial. The source reports computational evidence for related logarithmic root bounds but does not state a resolution.