Two-summand conjecture for polynomial images in upper triangular matrix algebras

Let $K$ be an algebraically closed field, let $p\in K[F_n]$ be a polynomial in noncommutative variables, and let $T_m(K)^{(t-1)}$ denote the relevant $(t-1)$st derived subspace of the upper triangular matrix algebra. Suppose that $\operatorname{ord}(p)=t$ with

$$1<t<m-1.$$

Two-summand conjecture. Then

$$p\bigl(T_m(K)\bigr)+p\bigl(T_m(K)\bigr)=T_m(K)^{(t-1)}.$$

The preceding theorem establishes that the image is contained in $T_m(K)^{(t-1)}$ and that every element of this space is a sum of some bounded number of image elements. The authors conjecture that two summands always suffice for $m\geq 5$ and $2\leq\operatorname{ord}(p)\leq m-2$; the claim remains open in the supplied text.