Algebraic versus span 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 $\lambda_0$, let $\mathop{\rm am}\nolimits_{\mathbf{t},T}(\lambda_0)$ be the multiplicity of $\lambda_0$ as a root of the $\mathbf{t}$-characteristic polynomial, and let $\mathop{\rm sm}\nolimits_{\mathbf{t},T}(\lambda_0)$ be the dimension of the linear span of those eigenvectors. Algebraic versus span multiplicity conjecture. If $\lambda_0\in\mathbb{C}$, then

$$\mathop{\rm am}\nolimits_{\mathbf{t},T}(\lambda_0)\ge\mathop{\rm sm}\nolimits_{\mathbf{t},T}(\lambda_0).$$

This conjecture generalizes the matrix inequality between geometric and algebraic multiplicities to tensor eigenvalues, for which the relations among algebraic, geometric, and span multiplicities are not fully understood. Its status is not established in the supplied text.