Two-sided Curto–Herrero conjecture for tuples of matrices

Let $M_n^m$ be the space of $m$-tuples of $n\times n$ matrices over a field, and let ${\rm GL}_n$ act by simultaneous conjugation. For ${\underline{A}}=(A_1,\ldots,A_m)$, write $f({\underline{A}})$ for the evaluation of a noncommutative polynomial $f$ in the tuple.

Two-sided Curto–Herrero conjecture. For every ${\underline{A}},{\underline{B}}\in M_n^m$, the ${\rm GL}_n$-orbits of ${\underline{A}}$ and ${\underline{B}}$ coincide if and only if

$$\operatorname{rank} f({\underline{A}})=\operatorname{rank} f({\underline{B}})$$

for every noncommutative polynomial $f$ in $m$ variables, possibly with a free term.

The conjecture is refuted in general for $n\geq3$ and $m\geq2$ by a counterexample of Hadwin and Larson. It does hold for $m=1$, by the rational canonical form; related rank criteria using matrices of noncommutative polynomials are captured by the solved two-sided Hadwin–Larson conjecture.