Duval–Reiner conjecture for simplicial-complex Laplacians

Let $K$ be a $k$-family on vertex set $V(K)$. Let $\partial_{k-1}$ be its boundary operator, and let $\mathbf{s}=(s_1\ge s_2\ge\cdots)$ be the nonincreasing sequence of nonzero eigenvalues of $\partial_{k-1}\partial_{k-1}^*$. Let $\mathbf{d}=(d_1\ge d_2\ge\cdots)$ be the vertex degree sequence of $K$, and let $\mathbf{d}^\top$ be its conjugate sequence. For nonincreasing sequences $\mathbf{x}$ and $\mathbf{y}$, $\mathbf{x}\prec\mathbf{y}$ means that their partial sums satisfy the majorization inequalities and their total sums are equal.

Duval–Reiner conjecture. The eigenvalue sequence is majorized by the conjugate vertex degree sequence:

$$\mathbf{s}\prec\mathbf{d}^\top.$$

This is proposed as a far-reaching analogue of the Grone–Merris conjecture for simplicial complexes. The supplied text does not state whether the conjecture has been proved or refuted.