The second-order asymptotic conjecture for ring-graph jigsaw percolation

Let $p_c(n)$ denote the critical probability for jigsaw percolation on the ring puzzle graph with an Erdős–Rényi people graph. Second-order asymptotic conjecture. There exist constants $b>0$, $c_1>0$, and $c_2$ such that

$$p_c(n)=\frac{c_1}{\log n}+\frac{c_2}{(\log n)^{1+b}}+o\left((\log n)^{-1-b}\right).$$

The authors motivate this as a refinement of the expected slow convergence of $p_c(n)$ to its critical asymptotic scale; the statement is presented as a conjecture, with no resolution supplied in the source.