Feasibility of partial alignment in the Erdős–Rényi model above the threshold

Let $p$ be the number of graphs, let $\boldsymbol{\boldsymbol{\pi}}^*$ be the underlying permutation tuple, and let $ov(\hat{\boldsymbol{\pi}},\boldsymbol{\pi}^*)$ denote their overlap. In the Erdős–Rényi model, with parameters $\lambda$ and $s$, assume

$$\lambda s\bigl(1-(1-s)^{p-1}\bigr)>1.$$

Feasibility conjecture. Partial alignment is feasible: there exists an estimator $\hat{\boldsymbol{\pi}}$ and some $\varepsilon>0$ such that, with high probability,

$$ov(\hat{\boldsymbol{\pi}},\boldsymbol{\pi}^*)>\varepsilon.$$

This statement complements the preceding theorem asserting intractability below the threshold $\lambda s(1-(1-s)^{p-1})<1$. The source presents the above-threshold claim without a proof or an explicit resolution, so its status remains open.