Reductive symmetry conjecture for recursively composed quasi-means

Let $G_2(A,B)=A\operatorname{\#} B$ be the two-variable geometric mean, and let a quasi-mean be obtained by recursively composing copies of $G_2$. A symmetry of such a quasi-mean is an invariance under a permutation of its matrix arguments.

Reductive symmetry conjecture. All the symmetries of a quasi-mean obtained by recursive composition from $G_2$ are reductive symmetries, meaning that each can be formally proved using only the reductive symmetries of the constituent copies of $G_2$.

This conjecture concerns which permutation symmetries can arise from recursive constructions of matrix geometric means. The source provides no resolution, so the conjecture remains open.