The sharp-threshold conjecture for ring-graph jigsaw percolation

For the ring puzzle, let $p_{\epsilon}(n)$ be the unique value of $p$ such that the probability that the jigsaw process solves the puzzle is $f0a0\epsilon$, for fixed $\epsilon\in(0,1)$. Sharp-threshold conjecture. For every fixed $\epsilon\in(0,1)$,

$$\frac{p_{\epsilon}(n)}{p_{1-\epsilon}(n)}\to 1$$

as $n\to\infty$. This asserts that the phase transition at the critical probability is asymptotically sharp, but the source presents it as an unresolved conjecture.