Fan's Laplacian multiplicity conjecture for connected uniform hypergraphs

Let $H$ be a connected $k$-uniform hypergraph, and let $\rho$ be the spectral radius of its adjacency tensor $\mathcal{A}(H)$. Let $\mathcal{L}(H)$ denote its Laplacian tensor. The projective eigenvariety $\mathbb{V}_{\lambda}(\mathcal{L}(H))$ is the set of eigenvectors associated with $\lambda$, considered in complex projective space. Fan's conjecture.

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

Here ${\rm am}(\lambda,\mathcal{A})$ denotes algebraic multiplicity. Fan proved related equalities for certain classes, including uniform hypertrees; the conjecture extends the asserted multiplicity equality to all connected uniform hypergraphs.