Volčič's invariant-subspace Nullstellensatz conjecture

Let $\mathbb{K}$ be a field, let $f_1,\ldots,f_l,g\in\mathbb{K}\langle\underline{x}\rangle$ be noncommutative polynomials, and for each $n\in\mathbb{N}$ and $\underline{X}\in\mathrm{M}_n(\mathbb{K})^d$ consider the joint invariant subspaces of $f_1(\underline{X}),\ldots,f_l(\underline{X})$. Volčič's conjecture. The following conditions are equivalent: (i) for all $n\in\mathbb{N}$ and $\underline{X}\in\mathrm{M}_n(\mathbb{K})^d$, every joint invariant subspace for $f_1(\underline{X}),\ldots,f_l(\underline{X})$ is invariant for $g(\underline{X})$; (ii) $g$ belongs to the unital $\mathbb{K}$-algebra generated by $f_1,\ldots,f_l$. This extends the directional-zero Nullstellensatz from joint kernel spaces to joint invariant subspaces. The conjecture is true for $l=1$, but remains open for $l>1$; the paper addresses the latter case.