Polylogarithmic gaps conjecture for the 6-GP-free process

Let $T$ be the sequence generated by the $6$-GP-free process, and interpret a gap of size $O((\log x)^C)$ as the existence of a constant implicit in the big-$O$ bound, independent of $x$. Polylogarithmic gaps conjecture. There exists a constant $C>0$ such that, with nonzero probability, the sequence $T$ has gaps of size

$$O((\log x)^C).$$

The conjecture proposes a substantial improvement over the paper's main bound and is motivated by the expected polylogarithmic scale of the largest gaps in the prime and squarefree sequences. It would give short gaps for a GP-free sequence while avoiding the unresolved distributional difficulties associated with those classical sequences.