Equidistribution conjecture for fractional parts of factorial roots

For a positive integer $A$, let

$$S=\left\{\sqrt[k]{a!}\bmod 1:2\leq a\leq A,\ 2\leq k\leq5\log\log A\right\},$$

where the set is counted as a multiset when cardinalities are computed. Equidistribution conjecture. For any interval $I\subset[0,1]$,

$$|S\cap I|=|S|\,|I|+O\left(|S|^{1-\epsilon}\right)$$

for some positive $\epsilon>0$ as $A\to\infty$. Such an estimate would provide the equidistribution needed to prove that the set of integers $a$ participating in solutions to $a!b!=c!$ has asymptotic density zero. The conjecture is not resolved in the source.