Uniqueness conjecture for two-sided quantum tangent spaces

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, take the span of its associated positive root vectors and regard it as a candidate quantum tangent space; a tangent space is right and left when it satisfies both the right- and left-tangent-space conditions. Uniqueness conjecture for two-sided quantum tangent spaces. The two reduced decompositions $\mathbf{j}$ and $\mathbf{j}'$ are the only reduced decompositions for which the span of the positive root vectors is both a right and a left tangent space. Low-rank computations show several distinct families according to whether the resulting space is two-sided, only left, only right, or neither; the asserted uniqueness beyond those computations remains open.