Characteristic-polynomial conjecture for parameter matrices of hypergraph transversals

Let $\mathbb{S}$ be the parameter matrix of a $k$-transversal in a $d$-uniform $r$-regular hypergraph $\mathcal{G}$. Let $\xi$ be a $d$-th primitive root of unity. Characteristic-polynomial conjecture. The characteristic polynomial of $\mathbb{S}$ is

$$\varphi(\lambda)=\lambda^{d-2}\prod\limits_{i=j}^d(\lambda-\xi^{jk}r).$$

The preceding result identifies the eigenvalues of the parameter matrix; this conjecture further specifies their algebraic multiplicities through the characteristic polynomial. The source also notes the belief that all nonzero eigenvalues have the same algebraic multiplicity, but provides no resolution of the conjecture.