Hu–Ye's eigenvariety bound for tensor eigenvalue multiplicity

Let $\mathcal{A}$ be a $k$-th order, $n$-dimensional tensor, let $\lambda$ be an eigenvalue, and let

$$\mathcal{V}_{\lambda}(\mathcal{A})=ccn$$

be its eigenvariety. Suppose that this variety has irreducible components $V_1,\ldots,V_\kappa$, and write $\mathrm{am}(\lambda)$ for the algebraic multiplicity and $\mathrm{gm}(\lambda)$ for the maximum dimension of an irreducible component. Hu–Ye's conjecture.

$$\mathrm{am}(\lambda) \geq \sum_{i=1}^{\kappa} \mathrm{dim}(V_i)(k-1)^{\mathrm{dim}(V_i)-1},$$

and, in particular,

$$\mathrm{am}(\lambda) \geq \mathrm{gm}(\lambda)(k-1)^{\mathrm{gm}(\lambda)-1}.$$

Hu and Ye proposed this bound in 2016 and proved it in several cases; the paper studies further cases, but the general assertion remains open.