Generic injectivity of recovery from coprime power sums

Let $n$ and $m$ be positive integers, let $a_1,\ldots,a_m$ be powers with $m=n+1$ and $\gcd(a_1,\ldots,a_m)=1$, and define

$$\phi(z_1,\ldots,z_n)=\left(\sum_{i=1}^n z_i^{a_1},\ldots,\sum_{i=1}^n z_i^{a_m}\right).$$

Generic-injectivity conjecture. The recovery of a set of $n$ complex numbers from $n+1$ power sums with coprime powers is unique: for generic $z\in\mathbb{C}^n$, the fiber $\phi^{-1}(\phi(z))$ consists exactly of the $n!$ coordinate permutations of $z$. This means that $\phi$ is generically injective modulo coordinate permutations. This would establish unique recovery from $n+1$ power sums in the coprime case; the source gives no resolution beyond stating the conjecture.