Kron-Knuth strengthening of the noncommutative super Schur formula

Let $\mathcal{A}$ be the ordered colored alphabet, let $\mathcal{U}$ be the associated noncommutative algebra, let $\mathfrak{J}_\nu(\mathbf{u})$ be the noncommutative super Schur function for a partition $\nu$, let $\mathrm{CT}_\nu^{<}$ denote the colored tableaux indexing the formula, and let $\operatorname{arwread}(T)$ be the associated reading word. Let $I_{\mathrm{Kron-K}}$ be the ideal generated by the stated colored plactic and Kronecker-Knuth relations. Kron-Knuth conjecture. For every partition $\nu$, one has

$$\mathfrak{J}_\nu(\mathbf{u})=\sum_{T\in\mathrm{CT}_\nu^{<}}\operatorname{arwread}(T)\quad\text{in }\mathcal{U}/I_{\mathrm{Kron-K}}.$$

This is presented as a strengthening of the paper's main theorem, and the stated computer checks cover the cases described in the surrounding text; the general assertion remains open.