Erdős–Graham–Granville–Selfridge conjecture on integral points on hyperelliptic curves

Let $t_n$ denote the quantity associated with the hyperelliptic equation in the paper, and let $c\in(0,1)$ be fixed. Assume that $n$ is a non-square integer sufficiently large in terms of $c$.

Erdős–Graham–Granville–Selfridge conjecture. One has

$$t_n\geq (\log n)^{1-c}.$$

This conjecture would strengthen the available lower bound for $t_n$ beyond the case where the associated hyperelliptic polynomial has even degree. It is motivated by the expectation that the bound $n\ll t_n^{O(t_n)}$ should also hold when the relevant number of factors is odd, and remains open in the stated generality.