The spectral gap conjecture for the \operatorname{Out}$F_n$ representation

Let $F_n$ be the free group of rank $n$, let $K$ be the compact group appearing in the paper, and let $\operatorname{Hom}(F_n,K)/K$ denote the quotient of the representation variety by conjugation. Write $L^2_0(\operatorname{Hom}(F_n,K)/K)$ for the orthogonal complement of the constant functions in $L^2(\operatorname{Hom}(F_n,K)/K)$. The spectral gap conjecture. The representation of $\operatorname{Out}(F_n)$ on

$$L^2_0(\operatorname{Hom}(F_n,K)/K)$$

has a spectral gap for $n>3$.

This conjecture concerns the large symmetry group of the free group and its action on the representation quotient. The source presents it as a natural conjecture in the context of its spectral-gap results; no resolution is given.