Generalized Kaplansky conjecture for simple finite-dimensional algebras

Let $F$ be an infinite field, and let $A$ be a simple finite-dimensional algebra over $F$. For a multilinear polynomial evaluated on $A$, consider the resulting set of values in $A$.

Generalized Kaplansky conjecture. The evaluation of any multilinear polynomial over $A$ is a vector subspace of $A$.

This extends the matrix-algebra question to simple finite-dimensional algebras. The source explains that the simplicity hypothesis is essential, since the assertion fails for the non-simple Grassmann algebra; the paper studies related Jordan-algebra cases.