Lusztig's conjecture on connectivity of Kazhdan–Lusztig right cells

Let $W$ be a Coxeter group, with Kazhdan–Lusztig right cells defined by the Kazhdan–Lusztig right preorder. A subset $C\subseteq W$ is right-connected if for every $x,y\in C$ there is a sequence $x=x_0,x_1,\dots,x_k=y$ in $C$ such that $x_i^{-1}x_{i-1}\in S$ for every $1\leq i\leq k$, where $S$ is the set of simple reflections. Lusztig's right-cell connectivity conjecture. Every Kazhdan–Lusztig right cell in a Coxeter group is right-connected. Lusztig formulated this for affine Weyl groups; it still appears to be open in finite type $D_n$ and in general affine type.