Duval–Reiner majorization conjecture for simplicial-complex Laplacians

Let $K$ be an $r$-dimensional simplicial complex. Using the spectrum $\bm{\lambda}_{r-1}(K)$ of the up-Laplacian and the conjugate vertex-degree sequence $\mathbf d_0^\top(K)$, with majorization defined by padded nonincreasing sequences, Duval–Reiner conjecture.

$$\bm{\lambda}_{r-1}(K) \preccurlyeq \mathbf d_0^\top(K).$$

Duval and Reiner proposed this higher-dimensional extension of the Grone–Merris–Bai theorem. The conjecture is refuted: the paper constructs pure complexes violating its fifth partial-sum inequality, while the second partial-sum inequality is confirmed and its equality case is characterized.