The Kazhdan–Lusztig–factorizable-sheaves equivalence

Let $G_1$ be a simple group, let $G_1$ be its Langlands dual group, and let $\kappa_1$ be the invariant form used to define the Kazhdan–Lusztig category $KL^{\kappa_1}(G_1)$. Let $FS^a(G_1)$ denote the factorizable-sheaves chiral category with parameter $a$, and let $d_1$ be the ratio of the squares of the lengths of the shortest and longest roots of $g_1$. Kazhdan–Lusztig–factorizable-sheaves conjecture. For $c_1\notin \mathbb Q$, there is an equivalence of chiral categories

$$KL^{\kappa_1}(G_1)\simeq FS^{\frac{1}{c_1d_1}}(G_1).$$

This equivalence is algebraic, meaning that it exists over an arbitrary ground field of characteristic $0$. The paper says that the conjecture is the subject of work in progress, and gives no resolution beyond that.