Upper bound on the maximal length of consecutive integers with distinct divisor counts

For a positive integer $m$, let $\tau(m)$ denote the number of positive divisors of $m$. For a real number $x \geq 1$, define $F(x)$ to be the maximum integer $k$ such that there exists a natural number $n$ with $n + k \leq x$ and the values $\tau(n+1), \tau(n+2), \ldots, \tau(n+k)$ are all distinct. Is it true that there exists a function $O: \mathbb{R} \to \mathbb{R}$ that is bounded in a neighborhood of infinity such that $F(x) \leq (\log x)^{O(x)}$ for all sufficiently large $x$?