Tripartite classification conjecture for connected derangement graphs

Let $G$ be a transitive permutation group of degree $n$, let $\rho(G)$ be its intersection density, and let $\Gamma_G$ be its derangement graph. Assume that $\Gamma_G$ is connected.

Tripartite classification conjecture. If

$$\rho(G)=\frac{n}{3},$$

then $\Gamma_G$ is complete tripartite.

The conjecture concerns the extremal case of the general upper bound $\rho(G)\leq n/3$. The source notes that complete tripartite derangement graphs are known in several examples, but does not provide a resolution of this classification claim.