Equality of the parking-function and Gelfand-pair polynomials

Let $\varGamma$ be a finite Abelian group, and for each $n\geq 1$ let $\widetilde{\varGamma^{n}}$ be the quotient of $\varGamma^n$ by its diagonal subgroup. In the cyclic case, define

$$C_n(q)=\sum_\alpha q^{\dim V_\alpha},$$

where $\bigoplus_\alpha V_\alpha$ is the decomposition into irreducibles of $\operatorname{Ind}^{\widetilde{\varGamma^{n}} \rtimes S_n}_{S_n}(1)$. Also, let $S(n)$ be the set of sequences $\mathbf{b}=(b_1,\dots,b_n)\in\mathbb{N}^n$ satisfying $\sum_{i=1}^j b_i\geq j$ for $j=1,\dots,n$ and $\sum_{i=1}^n b_i=n$, and define

$$S_n(q)=\sum_{\mathbf{b}\in S(n)}q^{\binom{n}{b_1,\ldots,b_n}}.$$

Equality conjecture. The polynomials $C_n(q)$ and $S_n(q)$ are identical for all $n\geq 1$. This would identify the dimension-generating polynomial arising from the multiplicity-free Gelfand-pair representation with the corresponding polynomial over parking-function sequences. The source gives no resolution or further evidence for this equality, so its status remains open.