Mesyan conjecture on images of multilinear polynomials

Let $K$ be a field, let $n\geq 2$ and $m\geq 1$ be integers, and let $f(x_1,\dots,x_m)$ be a nonzero multilinear polynomial in $K\langle x_1,\dots,x_m\rangle$. Mesyan conjecture. If $m\leq 2n-1$, then the image of $f$ on $M_n(K)$ contains $sl_n(K)$.

This is a weaker version of the Lvov-Kaplansky conjecture. The source reports that the conjecture is related to known results on polynomial images, but does not establish it in general.