Apte–Parekh–Sud token-excess matching conjecture

Let $G=(V,E)$ be a graph, let $F_k(G)$ be its $k$-th token graph, and define

$$\operatorname{\varepsilon}_k^T(G)=\lambda_1(L(F_k(G)))-|E|.$$

Let $\nu(G)$ denote the matching number of $G$. Apte–Parekh–Sud's token-excess conjecture. For $1\le k\le |V|$,

$$\operatorname{\varepsilon}_k^T(G)\le \nu(G).$$

This conjecture is stated after the paper's bounds for token graphs; those bounds do not establish the proposed matching-number upper bound, and the supplied status gives no resolution.