Monotonicity conjecture for the largest upper-Laplacian eigenvalue

Let $X$ be a finite simplicial complex, let $csigma_{\max}(\partial_k)$ denote the largest singular value of its $k$-th boundary operator, and let $L_k^{\mathrm{up}}$ denote the $k$-dimensional upper Laplacian. Monotonicity conjecture. For every finite simplicial complex and every $k\geq 1$, one has

$$\sigma_{\max}(\partial_{k+1})\leq\sigma_{\max}(\partial_k);$$

equivalently, $\lambda_{\max}(L_k^{\mathrm{up}})$ is non-increasing in $k$. The conjecture proposes a uniform higher-dimensional extension of the comparison studied for graphs; its status is not determined by the supplied text.