Laplacian zero-eigenvalue multiplicity conjecture for connected uniform hypergraphs

Let $H$ be a connected $k$-uniform hypergraph, let $\mathcal{L}(H)$ be its Laplacian tensor, let $\mathcal{A}(H)$ be its adjacency tensor, and let $\rho$ be the spectral radius of $\mathcal{A}(H)$. Write $\mathrm{am}(0,\mathcal{L}(H))$ and $\mathrm{am}(\rho,\mathcal{A}(H))$ for the corresponding algebraic multiplicities, and let $\mathbb{V}_0(\mathcal{L}(H))$ denote the projective eigenvariety for the zero eigenvalue. Laplacian zero-eigenvalue multiplicity conjecture.

$$\mathrm{am}(0,\mathcal{L}(H))=\mathrm{am}(\rho,\mathcal{A}(H)),$$

and consequently

$$\mathrm{am}(0,\mathcal{L}(H))=|\mathbb{V}_0(\mathcal{L}(H))|.$$

The first equality is established in the paper for $k$-uniform hypertrees, and the authors propose its extension to all connected $k$-uniform hypergraphs; the consequence then follows from the relation between the corresponding projective eigenvarieties.