Profinite semidistributivity of extended weak order

Prove that for every Coxeter group W, the extended weak order Dyer(W) is a profinite semidistributive lattice.

Background

The paper proves that the weak orders on all total orders of the integers, translation-invariant total orders, and the extended weak order of the affine symmetric group are profinite semidistributive lattices. It also establishes that profinite lattices have strong algebraic and semidistributive properties, including canonical join representations for appropriate elements.

These results motivate the proposed generalization from the affine symmetric group to arbitrary Coxeter groups. The conjecture asserts both profiniteness and semidistributivity of the extended weak order Dyer(W), and is not resolved in the paper for general Coxeter groups.