Weighted generic injectivity from power sums

Let $\tau=(\tau_1,\ldots,\tau_n)\in\mathbb{R}^n_{>0}$, and define

$$\psi:\mathbb{C}^n\longrightarrow\mathbb{C}^m,\qquad \psi_j=\sum_{i=1}^n\tau_i x_i^{a_j}.$$

Let $\operatorname{Stab}(\tau)$ be the subgroup of $S_n$ consisting of coordinate permutations that fix $\tau$. Weighted injectivity conjecture. For generic $z\in\mathbb{C}^n$, the fiber $\psi^{-1}(\psi(z))$ is precisely the set of all coordinate permutations of $z$, and its cardinality is $|\operatorname{Stab}(\tau)|$. This is a weighted generalization of the unweighted recovery problem; the source gives no resolution.