Weakly supercritical circumference conjecture for random planar graphs

Let $\mathcal P$ be the class of planar graphs, let $P=P(n,m)\in_R\mathcal P(n,m)$ be chosen uniformly at random, and let $L_1=L_1(P)$ be the largest component of $P$. Assume that

$$m=n/2+s,$$

where $s=s(n)=o(n)$ and $s^3n^{-2}\to\infty$. The circumference $c(L_1)$ is the length of the longest cycle in $L_1$.

Weakly supercritical circumference conjecture.

$$c(L_1)=\Theta_p\left(sn^{-1/3}\right).$$

This would sharpen the available lower and upper bounds for the circumference of the largest component in the weakly supercritical regime. The statement is presented as a consequence of the conjectured linear circumference of random cubic planar multigraphs.