Hibi's conjecture on integral lattices and Cohen–Macaulayness

Prove that every integral finite lattice is Cohen–Macaulay, thereby resolving Hibi's conjecture beyond the join-semidistributive and meet-semidistributive classes treated in the paper.

Background

A finite poset is called integral if it admits a homogeneous algebra with straightening laws that is a domain, while Cohen–Macaulayness is defined through the Stanley–Reisner ring of its order complex. Hibi established integrality for every distributive lattice and proposed that integrality should imply Cohen–Macaulayness for all finite lattices.

The paper proves this implication for join-semidistributive and meet-semidistributive lattices, including semidistributive lattices, but does not establish the conjecture for arbitrary finite lattices. Thus the general implication remains the unresolved problem explicitly identified by the authors.