Polylogarithmic non-giant components conjecture for the square graph

Let $[?]$ be an Erdős–Rényi random graph, $[?][?]$, and let $[?]([?])$ be the auxiliary square graph. Let $[?]>0$ be fixed, and write $[?]$ for a quantity bounded above and below by positive powers of $[?]$.

Polylogarithmic non-giant components conjecture. Suppose

$$p=p(n)\geq \frac{\sqrt{\sqrt{6}-2}+\varepsilon}{\sqrt{n}}.$$

Then a.a.s. every connected component of $[?]([?])$ except the largest has size $([?])^{O(1)}$.

This is a concluding open question about the component structure at and above the predicted giant-square-component threshold. The paper's theorem establishes a related polylogarithmic bound in a range beginning at a larger order, while this conjecture aims at the earlier threshold.