Saturated-valuation construction of unipotent bicrystals

Let $Y^\vee$ be an affine $G^\vee$-variety, let $S$ be a split algebraic torus, and let $\prec$ be a total ordering on the cocharacter lattice $X_\star(S)$. Assume there are a saturated valuation

$$\nu:\mathbb{C}[Y^\vee]\setminus\{0\}\to (X_\star(S),\prec)$$

and a perfect basis ${\bf B}$ for $\mathbb{C}[Y^\vee]$ such that the restriction of $\nu$ to ${\bf B}$ is injective. Then there exists a strongly positive $(U,\chi^{st})$-linear bicrystal $({\bf X},{\bf p},f,\Theta)$, with an isomorphism $\theta:S\overset\sim\to X^-$, for which the image of the valuation-induced crystal map is $\mathcal{B}_{f,\Theta}\subset X_\star(S)$ and induces an isomorphism

$$\mathcal{B}(\mathbb{C}[Y^\vee])\overset\sim\to \mathcal{B}_{f,\Theta}.$$

Saturated-valuation conjecture. Under these assumptions, such a strongly positive unipotent bicrystal exists and the valuation-induced map identifies the normal crystal of $\mathbb{C}[Y^\vee]$ with $\mathcal{B}_{f,\Theta}$.

The conjecture gives sufficient conditions for constructing strongly positive unipotent bicrystals from affine $G^\vee$-varieties, saturated valuations, and perfect bases. The source presents it as a route toward constructing bicrystals for suitable varieties; no resolution is stated.