The multivariate Abel–Ruffini conjecture for reduced polynomial systems

Let $A_1,A_2,\ldots,A_n\subset\mathbb{Z}^n$ be a reduced tuple, meaning that the dimension of the convex hull of $\bigcup_{j\in I}A_j$ is greater than $|I|$ for every $I\subsetneq\{1,2,\ldots,n\}$. Consider the general system of polynomial equations supported at $A_1,A_2,\ldots,A_n$. Multivariate Abel–Ruffini conjecture. The general system is solvable by radicals, equivalently in generalized quadratures, if and only if it has at most four solutions. This is the proposed multivariate analogue of the Abel–Ruffini theorem; the reduction hypothesis excludes systems whose solvability can be reduced to systems in fewer variables. The supplied text gives no resolution, so the conjecture remains open.