Inherited correspondence between disjoint cycles and non-zero eigenvalues

Let $G=([n],E)$ be a directed graph, and let $0 \leq k \leq n$ be an integer. For a subgraph $H$ of $G$, consider collections of disjoint cycles in $H$, and let $\mathcal{S}_G$ denote the associated matrix space.

Inherited correspondence conjecture. The maximum number of edges in a subgraph $H$ of $G$ in which every collection of disjoint cycles covers at most $k$ vertices equals the largest dimension of a subspace of $\mathcal{S}_G$ in which every matrix has at most $k$ non-zero eigenvalues.

This conjecture seeks to strengthen the established correspondence between collections of disjoint cycles and the non-zero eigenvalues of matrices in $\mathcal{S}_G$. Its proof or disproof is identified by the authors as an interesting open problem.