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

Let $T_m(K)$ denote the algebra of upper triangular $m\times m$ matrices over a field $K$, and let $p$ be a multilinear polynomial in noncommutative variables. Fagundes–de Mello conjecture. The image of $p$ on $T_m(K)$ is a vector space.

This is a variation of the Kaplansky–Lvov conjecture for upper triangular matrix algebras. The source says it was posed in 2019 and presents it as an open conjecture, with the surrounding discussion describing partial progress on related image problems.