Maximal-prolongation dimension conjecture for Lusztig root-vector calculi

Let $\mathbf{j}$ and $\mathbf{j}'$ be two reduced decompositions of the longest element of the Weyl group of $\mathfrak{sl}_{n+1}$. For each reduced decomposition, consider the associated space of positive root vectors when it forms a quantum tangent space, and let its associated full differential calculus have a maximal prolongation. Maximal-prolongation dimension conjecture. The two reduced decompositions $\mathbf{j}$ and $\mathbf{j}'$ are the only reduced decompositions for which the associated space of positive root vectors forms a tangent space whose associated full differential calculus has a maximal prolongation of classical dimension. The claim extends the observed low-order maximal-prolongation results and predicts that classical maximal-prolongation dimension singles out the Lusztig calculus and its dual; it remains open.