Cactus-to-toggle factorization conjecture for minuscule crystals

Let $\mathfrak{g}$ be a simple Lie algebra, let $\lambda$ be a minuscule dominant weight, and let $w_0^J$ be the corresponding Weyl group element. Write $C_{\mathfrak{g}}$ for the cactus group, $Tog(w_0^J)$ for the toggle group, and $RPP(w_0^J,n)$ for the reverse plane partitions associated with $w_0^J$. The crystal $B(n\lambda)$ is identified with $RPP(w_0^J,n)$. Cactus-to-toggle factorization conjecture. There is a surjective map

$$C_{\mathfrak{g}}\to Tog(w_0^J)$$

such that the action of the cactus group on the crystal

$$B(n\lambda)\cong RPP(w_0^J,n)$$

factors through this map. In type $D_m$, the action of the cactus group on the crystal $B(n\varpi_1)$ is generated by cactus elements corresponding to length $1$ and $2$ subdiagrams of the Dynkin diagram. This conjecture proposes a uniform relationship between cactus-group actions and toggle actions beyond type $A$, extending the known connection between Berenstein--Kirillov involutions and cactus groups in type $A$ and the toggle models for minuscule Demazure crystals in $ADE$ types. The status of the proposed factorization and the type $D_m$ generation statement is not established in the supplied text.