Least-eigenvalue conjecture for set-wise 3-intersecting perfect matchings

For $k\geq 11$, let $A_{[2k]}$, $A_{[2k-2,2]}$, and $A_{[2k-4,2,2]}$ be the class adjacency matrices in the perfect matchings association scheme, and define

$$\hat{B}_{3}(2k) = A_{[2k]}\frac{(k-3)(7k-10)}{30(2k-4)!!}+A_{[2k-2,2]}\frac{-2(k^{2}-10k+15)}{15(2k-4)!!}+A_{[2k-4,2,2]}\frac{2(k-5)}{5(2k-6)!!}.$$

Least-eigenvalue conjecture. The row sum and least eigenvalue of $\hat{B}_{3}(2k)$ are, respectively,

$$\frac{(2k-1)(2k-3)(2k-5)}{15}-1$$

and $-1$. Furthermore, the only modules with eigenvalue equal to $-1$ are $[2k-2,2]$, $[2k-4,4]$, and $[2k-6,6]$.

This conjecture is the spectral ingredient intended to establish the ratio bound with equality for set-wise $3$-intersecting perfect matchings. The paper verifies the required eigenvalue bounds for all modules except $[2k-6,4,2]$ and $[2k-6,2,2,2]$, so the remaining cases are open in the stated text.