The rank Nullstellensatz conjecture for noncommutative polynomials

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

Rank Nullstellensatz conjecture. The following are equivalent: (i) there is $K\in\mathbb{N}$ such that for every $n\in\mathbb{N}$ and every $\underline X\in\operatorname{M}_n(\mathbbm k)^d$,

$$\operatorname{rk}g(\underline X)\le K\cdot\max\{\operatorname{rk}f_1(\underline X),\dots,\operatorname{rk}f_\ell(\underline X)\};$$

(ii) $g\in(f_1,\dots,f_\ell)$.

This proposed strengthening would relate uniform rank bounds on matrix values to membership in the two-sided ideal generated by $f_1,\dots,f_\ell$. The source gives no resolution and presents it as a further assertion beyond Makar-Limanov’s conjecture.