Covering-number bound for Laplacian eigenvalue sums

Let $G=(V,E)$ be a graph, let $\tau(G)$ denote its covering number, and let $L(G)$ be the Laplacian matrix of $G$. For $1\leq i\leq |V|$, write $\lambda_i(L(G))$ for the $i$-th largest Laplacian eigenvalue. Covering-number conjecture. For all $\tau(G)\leq k\leq |V|$,

$$\sum_{i=1}^k \lambda_i(L(G)) \leq |E|+k\cdot\tau(G)-\binom{\tau(G)}{2}.$$

This conjecture proposes a sharper relation between Laplacian eigenvalue sums and the vertex covering number. The paper proves related bounds involving covering and matching numbers, but the displayed inequality is proposed as an open strengthening.