Conjecture on the lower bound for 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 $X(n,p)$, let $f_k(n)=n^{1/k(k+1)}(\log n)^{1/(k+1)}$, and let $\omega(n)$ be any function tending to infinity with $n$. Lower-bound conjecture. With high probability,

$$\frac{f_k(n)}{\omega(n)}\le M_k(n).$$

Together with the proved upper bound $M_k(n)\le f_k(n)\omega(n)$, this predicts the order of maximal persistence up to factors tending to infinity. The source presents this as an expected analogous lower bound, and does not report a resolution.