The critical-constant 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. Critical-constant conjecture. The leading constant in the asymptotic expansion of $p_c(n)$ satisfies

$$c_1=\frac{\pi^2}{6}.$$

This conjecture specifies the leading constant in the preceding proposed expansion for the critical probability; the source gives no resolution.