Higher nilpotency-class identity conjecture for unitriangular groups

Let $k\geq 11$ and let $\mathcal{H}\subset \mathsf{UT}(n,\mathbb{Q})$ be any finite set of matrices. Write $\mathcal{I}=\{1,2,\ldots,k+1\}$, let $\operatorname{PI}^{\mathcal{I}}$ be the associated map on words in the alphabet $\mathcal{I}$, and let $H_k$ and $\mathfrak{L}_{\geq d}(\log\mathcal{H})$ denote the polynomial and lower-degree Lie subspaces used in the paper. Higher nilpotency-class identity conjecture. There exist an integer $r\geq0$, positive rational numbers $\alpha_1,\ldots,\alpha_r$, and words $\boldsymbol{j}_s=j_{s,1}j_{s,2}\cdots j_{s,m_s}$ in $\mathcal{I}$, for $s=1,\ldots,r$, such that $\operatorname{PI}^{\mathcal{I}}(\boldsymbol{j}_s)\in\mathbb{Z}_{>0}\cdot(1,1,\ldots,1)$ and, for all matrices $B_1,\ldots,B_{k+1}\in\mathsf{UT}(n,\mathbb{Q})$ satisfying $\log B_i\in\mathfrak{L}_{\geq1}(\log\mathcal{H})$ and $\sum_{i=1}^{k+1}\log B_i\in\mathfrak{L}_{\geq2}(\log\mathcal{H})$,

$$\sum_{\sigma\in\operatorname{S}_{k+1}}H_k(\log B_{\sigma(1)},\ldots,\log B_{\sigma(k+1)})+\sum_{s=1}^r\alpha_s\sum_{\sigma\in\operatorname{S}_{k+1}}H_k(\log B_{\sigma(j_{s,1})},\ldots,\log B_{\sigma(j_{s,m_s})}) \in\mathfrak{L}_{\geq k+1}(\log\mathcal{H})+\mathfrak{L}_{\geq2}(\mathfrak{L}_{\geq2}(\log\mathcal{H})).$$

The conjecture would supply the identities needed to extend the paper's polynomial-time computation of invertible subsets from nilpotency class at most ten to higher nilpotency classes; the authors identify proving the corresponding proposition for $k\geq11$ as the remaining obstacle, and no resolution is given here.