Lew's higher-dimensional Brouwer Laplacian conjecture

Let $K$ be an $r$-dimensional simplicial complex, let $f_r(K)$ denote the number of $r$-faces, and let $\lambda_{r-1,i}(K)$ be the eigenvalues of the up-Laplacian on $(r-1)$-faces, arranged in nonincreasing order. Lew's conjecture. For every integer $\ell \geq 1$,

$$\sum_{i = 1}^{\ell}\lambda_{r-1,i}(K) \leq f_r(K) + \binom{\ell}{2} + r\ell.$$

This is proposed as a higher-dimensional analogue of Brouwer's Laplacian conjecture for graphs. The source does not report a resolution of Lew's conjecture.