Generic Galois group of idempotents

Prove that the Galois group of the non-zero idempotents of the generic n-dimensional evolution algebra over the rationals is the full symmetric group on $2^n-1$ letters, and derive the stated limiting distribution over finite fields.

Background

A perfect n-dimensional evolution algebra has an idempotent scheme of total length $2^n$, with the zero idempotent contributing one reduced point. The conjecture predicts maximal monodromy for the remaining $2^n-1$ idempotents and consequently a Poisson-type fixed-point distribution for random finite-field algebras as the field size grows.