Multiplicity–eigenvector-variety conjecture for connected uniform hypergraphs

Let $H$ be a connected $k$-uniform hypergraph, let $\mathcal{A}(H)$ be its adjacency tensor, let $\rho$ be its spectral radius, and let $\mathbb{V}_{\rho}(\mathcal{A}(H))$ be the projective eigenvariety associated with $\rho$. Write $\mathrm{am}(\rho)$ for the algebraic multiplicity of $\rho$. Multiplicity–eigenvector-variety conjecture.

$$\mathrm{am}(\rho)=\left|\mathbb{V}_{\rho}(\mathcal{A}(H))\right|.$$

The equality is proved in the paper for hypertrees, powers of connected simple graphs, and complete $3$-uniform hypergraphs on at least four vertices; it is conjectured for all connected $k$-uniform hypergraphs.