Finite-fiber conjecture for characteristic polynomials of symmetric tensors

Let $n\geq 1$ and $d\geq 3$. Let $\Phi_{\mathrm{sym}}$ be the restriction of the characteristic polynomial map to the space of symmetric tensors:

$$\Phi_{\mathrm{sym}}:S^d\mathbb{C}^{n+1}\longrightarrow \mathbb{C}[\lambda]^{\mathit{mon}}_{D(n,d)}.$$

Finite-fiber conjecture for symmetric tensors. All fibers of $\Phi_{\mathrm{sym}}$ are finite.

In the symmetric setting, the analogue of the symmetry group acting on partially symmetric tensors is finite, motivating algebraic identifiability of the characteristic polynomial map. The source gives no resolution of this conjecture.