The finite-cover topological overlap conjecture

Let $X$ be a connected $2$-dimensional simplicial complex whose fundamental group has property T. The finite-cover topological overlap conjecture. There exists $c>0$ such that every finite cover of $X$ has the $c$-topological overlap property. This is proposed as a consequence of higher-property-T methods together with integral and mod-$p$ expansion; the source gives no resolution status.