Multiplicity binomiality conjecture for Burnside-chain eigenvalues

Fix $k$, and let $\lambda$ be any nonzero eigenvalue of the Burnside chain on $(C_k^n,S_n)$ for any $n$. Multiplicity binomiality conjecture. The eigenvalue $\lambda$ occurs with multiplicity

$$a_\lambda\binom{n}{b_\lambda}$$

for some integers $a_\lambda,b_\lambda$. This conjecture is motivated by the Schur--Weyl decomposition and seeks a uniform description of eigenvalue multiplicities as $n$ varies; proving it would help provide matching upper bounds for mixing time.