Smallest spectral-gap conjecture for perfect matching association scheme graphs

Let $n\geqslant 5$, and let $A_\mu$ be the adjacency matrix of the relation indexed by a partition $\mu\vdash n$ in the perfect matching association scheme. Exclude the trivial relation indexed by $[1^n]$.

Smallest spectral-gap conjecture. Among the adjacency matrices

$$\{A_\mu\mid \mu\vdash n,\ \mu\neq[1^n]\},$$

the smallest spectral gap is attained by $A_{[2,1^{n-2}]}$.

The spectral gap controls connectivity-related properties of these regular graphs, and the conjecture identifies the relation expected to minimize it among all nontrivial relations. The source gives no evidence of a resolution, so the conjecture remains open.