Lubotzky–Weiss conjecture on amenable and property (T) dense subgroups

Let $K$ be a compact group, and let $\Lambda_1$ and $\Lambda_2$ be finitely generated subgroups dense in $K$. Lubotzky–Weiss conjecture. If $\Lambda_1$ is amenable and $\Lambda_2$ has Kazhdan's property $(\mathrm{T})$, then $K$ is finite.

The conjecture asks whether these opposing properties of two finitely generated dense subgroups force the ambient compact group to be finite. It was resolved in the negative by Ershov and Jaikin-Zapirain, and independently by Kassabov around 2010; the paper constructs further extreme counterexamples.