Largest-diameter conjecture for perfect matching association scheme graphs

Let $X_\mu$ denote the graph in the perfect matching association scheme associated with the relation indexed by a partition $\mu$ of $n$. In particular, consider the graph $X_{[2,1^{n-2}]}$.

Largest-diameter conjecture. The diameter of $X_{[2,1^{n-2}]}$ is the largest diameter among all graphs in the perfect matching association scheme.

The graph $X_{[2,1^{n-2}]}$ is known to have diameter $n-1$, and the conjecture is motivated by the relationship between spectral gap and diameter. The source explicitly presents the claim as a conjecture and does not state that it has been resolved.