Degree-refined 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 call $E_1,\dots,E_s$ its irreducible components. Let $\deg(E_i)$ and $\dim(E_i)$ denote their degrees and dimensions, and let $\mathop{\rm am}\nolimits_{\mathbf{t},T}(\lambda_0)$ be the algebraic multiplicity. Degree-refined algebraic versus geometric multiplicity conjecture. For $\lambda_0\in\mathbb{C}$,

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

This is presented as a stronger refinement of the algebraic-versus-geometric multiplicity conjecture because it incorporates the scheme-theoretic degrees of the irreducible components. The supplied text gives no general resolution.