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 such that $f(n)\to \infty$ as $n\to \infty$, then is $F(f,n)$ everywhere dense?