Tropical tensor-product factorization of Kashiwara crystals

Let $F$ be the positive map referenced in the source, and let $\mathcal{B}_{f_Z,\Theta_Z;\lambda,\nu}$ be the fiber of the invariant projection associated with $\mathcal{Z}_{w_0}$ over $(\lambda,\nu)\in X_\star(T)^+\times X_\star(T)^+$. Then:

Tropical tensor-product conjecture. The tropicalization of $F$ is an isomorphism of torsion-free Kashiwara crystals

$$\mathcal{B}_{\Theta^-_B}\times\mathcal{B}_{\Theta^-_B}\to\mathcal{B}_{f_Z,\Theta_Z},$$

and its restriction is an isomorphism of normal Kashiwara crystals

$$\tilde {\bf F}_{w_0}:\mathcal{B}_{f_B,\Theta^-_B}\times\mathcal{B}_{f_B,\Theta^-_B}\to\mathcal{B}_{f_Z,\Theta_Z}.$$

For every $\lambda,\nu\in X_\star(T)^+$, the further restriction gives

$$\mathcal{B}(V_\lambda)\times\mathcal{B}(V_\nu)\overset\sim\to\mathcal{B}_{f_Z,\Theta_Z;\lambda,\nu}.$$

This is presented as a refinement implied by the geometric tensor-product conjecture. The preceding lemma describes the fibers by tensor-product multiplicities, but the asserted crystal isomorphisms remain conjectural in the source.