Apte–Parekh–Sud token-graph Laplacian conjecture

Let $G=(V,E)$ be a graph and let $F_k(G)$ be its $k$-th token graph, whose vertices are the $k$-element subsets of $V$. For $1\le k\le |V|/2$, let $L(F_k(G))$ denote the Laplacian matrix of $F_k(G)$, and let $\lambda_1$ denote its largest eigenvalue. Apte–Parekh–Sud's token-graph conjecture. One has

$$\lambda_1(L(F_k(G)))\le |E|+k.$$

This is described as a token-graph analogue of Brouwer's conjecture and is motivated by approximation ratios for certain quantum algorithms. The paper proves the weaker bound $\lambda_1(L(F_k(G)))\le |E|+4k-2$, and the supplied status does not resolve the conjecture.