Kaplansky–Lvov multilinear conjecture for matrix algebras

Let $f$ be a multilinear polynomial with complex coefficients, and let $d\in\mathbb{N}$. Consider the set of values obtained by evaluating $f$ on $M_d(\mathbb{C})$. Kaplansky–Lvov multilinear conjecture. The set of values of $f$ in $M_d(\mathbb{C})$ is a vector space. This is a fundamental open problem about images of noncommutative polynomials in matrix algebras. The paper establishes the conjecture for degree-three polynomials when $d$ is even or $d<17$, but the general multilinear case remains open.