The crystal-structure conjecture for reverse King tableaux

Let $B(\lambda)$ be a highest weight crystal with highest weight element $u_{\lambda}$, and fix $w\in W$ with a reduced expression $w=s_{i_1}\cdots s_{i_{\ell}}$. Define the Demazure crystal by

$$B_w(\lambda):=\{b\in B(\lambda)\mid e_{i_{\ell}}^{a_{\ell}}\cdots e_{i_2}^{a_2}e_{i_1}^{a_1}b=u_{\lambda}\text{ for some }a_1,\ldots,a_{\ell}\in\mathbb Z_{\geq 0}\},$$

and define the corresponding atom crystal by

$$\overline{B}_w(\lambda):=B_w(\lambda)\setminus\bigcup_{v<w}B_v(\lambda).$$

Let $\mathcal{T}_{\lambda}$ be the set of reverse King tableaux of shape $\lambda$, let $\mathcal{K}_{\lambda}$ be the corresponding King tableaux, and let $\operatorname{key}$ denote the lattice-model key map.

Crystal-structure conjecture. There exists a crystal structure on $\mathcal{T}_{\lambda}$ such that

$$\{T\in\mathcal{T}_{\lambda}\mid \operatorname{key}(T)=w\}=\overline{B}_w(\lambda).$$

Additionally, tableau switching $\mathcal{T}_{\lambda}\to\mathcal{K}_{\lambda}$ is a crystal isomorphism with the crystal structure on $\mathcal{K}_{\lambda}$ given by Lee. Moreover, the composition of tableau switching interchanging $i\leftrightarrow\overline{\imath}$ for all $i$, entrywise replacement $i\leftrightarrow\overline{n+1-i}$, and the Sheats bijection is a crystal isomorphism on the corresponding crystal atoms and sends $\operatorname{key}$ to the key map defined by Joseph–Lam and Santos.

This conjecture would make the key map natural from the perspective of crystal theory. It connects reverse King tableaux, King tableaux, Kashiwara–Nakashima tableaux, DeConcini tableaux, and their corresponding crystal atoms. The source gives no evidence that the conjecture has been resolved.