Panja–Prasad's two-summand conjecture for upper triangular matrix algebras

Let $K$ be an algebraically closed field, let $n\geq 2$ and $m\geq 1$ be integers, and let $T_n(K)$ denote the algebra of $n\times n$ upper triangular matrices over $K$. For $t\geq 0$, let $T_n(K)^{(t)}$ consist of the upper triangular matrices whose $(i,j)$-entries are zero whenever $j-i\leq t$. Let $p(x_1,\ldots,x_m)$ be a noncommutative polynomial with zero constant term, and let $\operatorname{ord}(p)$ be its order, namely the least positive integer $r$ such that $p(T_r(K))=\{0\}$ but $p(T_{r+1}(K))\neq\{0\}$. Suppose that $\operatorname{ord}(p)=r$ with $1<r<n-1$. Panja–Prasad's conjecture.

$$p(T_n(K))+p(T_n(K))=T_n(K)^{(r-1)}.$$

The preceding result gives the inclusion $p(T_n(K))\subseteq T_n(K)^{(r-1)}$ and shows that every element of $T_n(K)^{(r-1)}$ is a sum of finitely many elements from $p(T_n(K))$; the conjecture asserts that two summands always suffice. Its status is not resolved in the supplied source.