Conjecture on the limiting distribution of maximal persistence in random clique complexes

Let $X(n,p)$ be the random clique complex, let $M_k(n)$ denote the maximal persistence over all $k$-dimensional cycles in $X(n,p)$), and let

$$f_k(n)=n^{1/k(k+1)}(\log n)^{1/(k+1)}.$$

Limiting-distribution conjecture. The random variable

$$\frac{M_k(n)}{f_k(n)}$$

converges in law to a limiting distribution supported on an interval $[\lambda_k,\infty)$ for some $\lambda_k>0$. Such a limit theorem would give precise probabilities for the existence of cycles whose persistence exceeds a specified threshold. The source offers this as a conjecture illustrated by numerical experiments and gives no resolution.