Characteristic polynomial fiber-dimension conjecture for partially symmetric tensors

Let $n \geq 2$ and $d \geq 3$, with $(n,d) \notin\\{(2,3),(3,3),(2,4)\\}$. Let $\Phi$ be the characteristic polynomial map on partially symmetric tensors

$$\Phi:\mathbb{C}^{n+1}\otimes S^{d-1}\mathbb{C}^{n+1}\longrightarrow \mathbb{C}[\lambda]^{\mathit{mon}}_{D(n,d)},\qquad T\longmapsto \varphi_T(\lambda).$$

Characteristic polynomial fiber-dimension conjecture. The generic fiber of $\Phi$ is equidimensional of dimension $n$.

The conjecture refines an earlier conjecture that the generic fiber is finite, which is false because a subgroup $H$ preserving the characteristic polynomial acts on the tensor space. The excluded cases are among those where $\Phi$ is dominant, and the dimension $n$ is known to be a lower bound for the generic fiber dimension; the asserted equidimensionality remains the proposed statement.