The invariant-subspace certificate conjecture for noncommutative polynomials

Let $\mathbbm k$ be the base field, let $\mathbbm k\!\mathop{<}\!\underline{x}\!\mathop{>}$ be the free algebra, and let $f_1,\dots,f_\ell,g\in\mathbbm k\!\mathop{<}\!\underline{x}\!\mathop{>}$.

Invariant-subspace certificate conjecture. The following are equivalent: (i) for every $n\in\mathbb{N}$ and every $\underline X\in\operatorname{M}_n(\mathbbm k)^d$, every joint invariant subspace for $f_1(\underline X),\dots,f_\ell(\underline X)$ is invariant for $g(\underline X)$; (ii) $g$ belongs to the unital $\mathbbm k$-algebra generated by $f_1,\dots,f_\ell$.

This is proposed as an analogue of Bergman’s algebraic certificate for inclusion of joint kernels, extending the geometric Nullstellensatz perspective from zeros to joint invariant subspaces. The source does not state a resolution.