Strong Mesyan-type conjecture for multilinear polynomial images

Let $K$ be a field, let $n\geq 2$ be an integer, and let $f(x_1,\dots,x_m)$ be a multilinear polynomial. Say that $f$ satisfies $(S1)$ when it is a central polynomial, so its image is contained in the scalar matrices $K$. Write $Im(f)$ for the image of $f$ on $M_n(K)$, and let $sl_n(K)$ denote the trace-zero matrices. Strong Mesyan-type conjecture. If $f$ does not satisfy $(S1)$, then

$$Im(f)\supseteq sl_n(K).$$

The source states that this conjecture is stronger than Mesyan's conjecture and weaker than the Lvov-Kaplansky conjecture; no general resolution is given.