Gamburd–Jakobson–Sarnak spectral-gap conjecture for generic compact-group generators

Let $G$ be a compact simple Lie group, let $(g_1,\ldots,g_k)\in G^k$ be a random $k$-tuple, and let $\Gamma(g_1,\ldots,g_k)<G$ be the subgroup it generates. Gamburd–Jakobson–Sarnak conjecture. For $k\geqslant 2$, for almost every $k$-tuple, the action

$$\Gamma(g_1,\ldots,g_k)\curvearrowright G$$

by left multiplication has a spectral gap. This conjecture predicts generic spectral-gap behavior for actions of subgroups generated by random elements of compact simple Lie groups; related results are known for several compact groups, but the general assertion is presented as open.