The binomial multiplicity conjecture for the Burnside chain

Fix $k$, and consider the Burnside chain on $(C_k^n,S_n)$ for varying $n$. Let $\lambda$ be any nonzero eigenvalue of this chain. The binomial multiplicity conjecture. There exist integers $a_\lambda$ and $b_\lambda$ such that the multiplicity of $\lambda$ is

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

The paper states this as a conjectural pattern while noting that a full description of the eigenvalues for the general $(C_k^n,S_n)$ Burnside process is not known.