Pattern conjecture for eigenvalues in the perfect matchings association scheme

Consider the perfect matchings association scheme on $K_{2k}$, with classes indexed by partitions and modules indexed by partitions. For an integer $i$, let $[2k-2i,2i]$ and $[2k-2i,2,\ldots,2]$ denote the indicated modules.

Eigenvalue pattern conjecture. The eigenvalues of the class $[2k]$ corresponding to these modules are, respectively,

$$-(2i-3)!!(2k-2i-2)!!$$

and

$$(-1)^i(i!)(2k-2i-2)!!.$$

The statement records patterns observed in the character tables of the perfect matchings association scheme. The source presents them as further work rather than proving them, and gives no resolution beyond the displayed formulas.