Makar-Limanov's low-rank values conjecture for noncommutative polynomials

Let $\mathbbm k$ be the base field, let $\mathbbm k\!\mathop{<}\!\underline{x}\!\mathop{>}$ be the free algebra, and let $f$ be a nonconstant noncommutative polynomial, so $f\in\mathbbm k\!\mathop{<}\!\underline{x}\!\mathop{>}\setminus\mathbbm k$. For each $n\in\mathbb{N}$, write $f(\operatorname{M}_n(\mathbbm k)^d)$ for the set of values of $f$ on $d$-tuples of $n\times n$ matrices.

Makar-Limanov's conjecture.

$$\inf_{n\in\mathbb{N}}\frac{\min\operatorname{rk} f(\operatorname{M}_n(\mathbbm k)^d)}{n}=0.$$

The conjecture asserts that every nonconstant noncommutative polynomial takes values whose rank is arbitrarily small relative to the matrix size, despite the possibility that it has no matricial zeros. It was proposed by Makar-Limanov; the source presents the subsequent examples only as scarce evidence and does not state a resolution.