The Lusztig-cone region-of-linearity conjecture for type A canonical bases

Let $w_0$ be the longest element of the Weyl group, let $\mathbf{i}$ and $\mathbf{j}$ be reduced expressions of $w_0$, let $C_{\mathbf{i}}$ be the Lusztig cone, and let

$$S_{\mathbf{i}}^{\mathbf{j}}:Y_{\mathbf{i}}\rightarrow {\mathbb{N}}^k$$

be the transition function from the string cone $Y_{\mathbf{i}}$, with $R=R_{\mathbf{j}}^{\mathbf{j}'}$ Lusztig's piecewise-linear function. Lusztig-cone region-of-linearity conjecture. For any reduced expression $\mathbf{i}$ of $w_0$, $S_{\mathbf{i}}^{\mathbf{j}}(C_{\mathbf{i}})$ is a region of linearity of Lusztig's piecewise-linear function $R=R_{\mathbf{j}}^{\mathbf{j}'}$. The conjecture identifies the image of each Lusztig cone under the canonical-basis transition map with a linearity region of Lusztig's piecewise-linear parametrization, clarifying the relationship between Lusztig-cone and string-cone coordinates. Its resolution is not indicated in the supplied text.