Lew's matching-number conjecture for Laplacian eigenvalue sums

Let $G$ be a finite simple graph with $n$ non-isolated vertices. Let $L(G)$ be its Laplacian matrix, let $\lambda_1(G)\ge\cdots\ge\lambda_n(G)$ be the Laplacian eigenvalues, and write

$$s_k(G)=\sum_{i=1}^k\lambda_i(G).$$

Let $\nu(G)$ denote the matching number of $G$. Lew's matching-number conjecture. If $1\le k\le n-2$, then

$$s_k(G)\le e(G)+k\nu(G).$$

This conjecture refines parameter-dependent upper bounds for sums of the largest Laplacian eigenvalues by using the matching number. The supplied text does not state whether Lew's conjecture has been resolved.