Algebraic versus geometric multiplicity conjecture for tensor eigenvalues

Let $n,d$ be positive integers, let $T,\mathbf{t}\in\mathbb{C}^n\otimes((\mathbb{C}^n)^*)^{\otimes d}$, and assume that $\mathbf{t}$ is non-singular. For $\lambda_0\in\mathbb{C}$, let $E_{\mathbf{t},T}(\lambda_0)$ be the eigenscheme of $\mathbf{t}$-eigenvectors of $T$ with eigenvalue $\lambda_0$, and let $E_1,\dots,E_s$ be its irreducible components. Let $\mathop{\rm am}\nolimits_{\mathbf{t},T}(\lambda_0)$ denote the algebraic multiplicity and $\mathop{\rm gm}\nolimits_{\mathbf{t},T}(\lambda_0)$ the dimension of the eigenscheme. Algebraic versus geometric multiplicity conjecture.

$$\mathop{\rm am}\nolimits_{\mathbf{t},T}(\lambda_0)\ge\sum_{i=1}^s \dim(E_i)d^{\dim(E_i)-1}$$

and

$$\mathop{\rm am}\nolimits_{\mathbf{t},T}(\lambda_0)\ge \mathop{\rm gm}\nolimits_{\mathbf{t},T}(\lambda_0)d^{\mathop{\rm gm}\nolimits_{\mathbf{t},T}(\lambda_0)-1}.$$

The second inequality is weaker than the first because some irreducible component has dimension equal to the geometric multiplicity. The conjecture is known for some special families of tensors arising from hypergraphs, but remains open in general.