Kashiwara–Nakashima–Okado conjecture on positive geometric crystals and perfect crystals

Let $G$ and $\mathfrak g=\langle \mathfrak t,e_i,f_i\rangle_{i\in I}$ be the affine Kac–Moody group and algebra associated with a generalized Cartan matrix, and let $B^-_w$ be the Schubert-cell model defined by products of the elements $Y_i(c)$. For $i\in I$, let $W(\varpi_i)$ be the fundamental representation with extremal weight $\varpi_i$, let $\mathbb P(\varpi_i)$ be its projectivization, let $c_i^\vee={\rm \max}(1,2/(\alpha_i,\alpha_i))$, and for an integral weight $\mu$ set $I(\mu)=\{j\in I\mid\langle\alpha_j^\vee,\mu\rangle\geq0\}$. For an extremal vector $u\in W(\varpi_i)_\mu$, write $t(c_i^\vee\mu)=\iota w$ in the extended Weyl group, with $\iota$ a Dynkin diagram automorphism and $w=s_{i_1}\cdots s_{i_k}$. Kashiwara–Nakashima–Okado conjecture. For every $i\in I$, there exist a unique variety $X$ endowed with a positive $\mathfrak g$-geometric crystal structure and a rational mapping $\pi\colon X\longrightarrow\mathbb P(\varpi_i)$ such that, for every such $u$, there is a birational mapping $\xi\colon B^-_w\longrightarrow X$ which is a morphism of $\mathfrak g_{I(\mu)}$-geometric crystals and for which $\pi\circ\xi$ maps $Y_{i_1}(x_1)\cdots Y_{i_k}(x_k)$ to $Y_{i_1}(x_1)\cdots Y_{i_k}(x_k)\overline u$, where $\overline u$ is the line containing $u$; moreover, the ultra-discretization of $X$ is isomorphic to the crystal $B_\infty(\varpi_i)$ of the Langlands dual $\mathfrak g^L$. The conjecture is known in several cases, including $i=1$ for the listed affine Lie algebras, while other cases remain open.