Polylogarithmic size of the second-largest square components

Prove that if $\Gamma\sim\mathcal{G}_{n,p}$ with $p\geq(\sqrt{\sqrt{6}-2}+\varepsilon)/\sqrt n$, then asymptotically almost surely every connected component of $T_1(\Gamma)$ other than the largest has size bounded by a fixed power of $\log n$.

Background

The paper proves that above a larger threshold there is a unique giant component in $T_1(\Gamma)$ and that all other components have size $O((\log n/\log\log n)^2)$. The conjecture asks whether the same qualitative conclusion already holds at the threshold where the giant square component first emerges.

Here $T_1(\Gamma)$ has non-edges of $\Gamma$ as vertices, with adjacency determined by induced squares in $\Gamma$; the conjecture concerns the sizes of all non-giant square components.