Injectivity on connected components of positive unipotent bicrystals

Let $({\bf X},{\bf p},f,\Theta)$ be a positive $(U\times U,\chi^{st})$-linear bicrystal of type $w$. The structure map

$$\tilde {\bf f}_w:\mathcal{B}_\Theta\to \mathcal{B}_{\Theta_{B^-}}$$

restricts injectively to each connected component of $\mathcal{B}_{f,\Theta}$.

Injectivity conjecture. For every connected component of $\mathcal{B}_{f,\Theta}$, the restriction of $\tilde {\bf f}_w$ is injective.

Informally, this predicts that the connected components of the tropical crystal reflect $U\times U$-orbit data, or equivalently classes of orbits in $X^-$ under the rational $U$-action. The conjecture is presented as a complement to the results of the section; no resolution is given in the source.