Stauduhar's conjecture on the number of distinct factorial residues

Let $p$ be a prime and let $h(p)$ denote the number of distinct residues of $1!,2!,\dots,(p-1)!$ modulo $p$.

Stauduhar's conjecture.

$$\lim_{p\to\infty}\frac{h(p)}{p}=1-\frac{1}{e}.$$

This conjecture predicts that the factorial sequence misses about $p/e$ residue classes modulo $p$. It remains unsolved; the asserted limit is known in average for modular mappings and is supported by numerical calculations.