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

The statement

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 $x$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.