The Duval–Reiner conjecture for family-pair Laplacian spectra

Let $(\mathcal K,\mathcal K')$ be a family pair, with relative Laplacian spectrum $\mathbf{s}(\mathcal K,\mathcal K')$ and transposed relative generalized degree sequence $\mathbf{d}(\mathcal K,\mathcal K')^T$. Family-pair majorization conjecture.

$$\mathbf{s}(\mathcal K,\mathcal K')\trianglelefteq\mathbf{d}(\mathcal K,\mathcal K')^T.$$

This extends the Grone–Merris and Duval–Reiner majorization conjectures from families to family pairs. The text says it is based on only a few examples and does not state a resolution.