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on Weisenberg's variant of Erdős problem 891 on prime factor counts in short intervals

The statement

Let $p_1, p_2, \ldots$ denote the sequence of prime numbers in increasing order, so that $p_1 = 2$ . For an integer $m \geq 1$ , let $\omega(m)$ denote the number of distinct prime factors of $m$ . For an integer $k \geq 2$ , let $P_k = p_1 p_2 \cdots p_k = \prod_{i=0}^{k-1} p_{i+1}$ . Determine whether there exist infinitely many positive integers $n$ such that every integer $m$ in the interval $[n, n + P_k - 1)$ satisfies $\omega(m) \leq k$ .

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.