OPEN

If $\tau(n)$ counts the number of divisors of $n$ then let
\[F(f,n)=\frac{\tau((n+\lfloor f(n)\rfloor)!)}{\tau(n!)}.\]
Is it true that
\[\lim_{n\to \infty}F((\log n)^C,n)=\infty\]
for large $C$? Is it true that $F(\log n,n)$ is everywhere dense in $(1,\infty)$? More generally, if $f(n)\leq \log n$ is a monotonic function then is $F(f,n)$ everywhere dense?

Erdős and Graham write that it is easy to show that $\lim F(n^{1/2},n)=\infty$, and in fact the $n^{1/2}$ can be replaced by $n^{1/2-c}$ for some small constant $c>0$.