Conjecture on Laplacian and signless Laplacian H-eigenvalues of even-uniform power hypergraphs

Let $G=(V,E)$ be an ordinary graph, let $k=2r$ be even, and let $G^k=(V^k,E^k)$ be the $k$-power hypergraph of $G$. Let $\mathcal L^k$ and $\mathcal Q^k$ be the Laplacian and signless Laplacian tensors of $G^k$, respectively. Even-uniform power-hypergraph conjecture. The common eigenvalues

$$\{\lambda(\mathcal L^k)=\lambda(\mathcal Q^k)\}$$

form a strictly decreasing sequence. This conjecture concerns the relationship between the Laplacian and signless Laplacian spectra of even-uniform power hypergraphs; the source provides no resolution, so its status remains open.