Fagundes–Mello conjecture on multilinear polynomial images of upper triangular matrix algebras

Let $K$ be a field and let $UT_n$ denote the algebra of all $n\times n$ upper triangular matrices over $K$. Let $f$ be a multilinear polynomial, and let $f(UT_n)$ denote its image under evaluation on $UT_n$. Fagundes–Mello conjecture. The image of $f$ over the field $K$ on $UT_n$ is always a vector space. The conjecture is presented as an important variation of the Lvov–Kaplansky conjecture. The source states that it remains unresolved and that the paper improves known results toward it.