Existence of shellable but not CL-shellable finite atomic lattices

Determine whether there exist finite atomic lattices that are shellable but not CL-shellable; equivalently, ascertain whether an lcm-lattice of a monomial ideal can be shellable without being CL-shellable.

Background

In general poset theory, CL-shellability implies shellability, but the converse fails: Vince and Wachs constructed a shellable poset that is not CL-shellable. However, their example is not a lattice. For lcm-lattices (equivalently, finite atomic lattices), no example is known that is shellable but not CL-shellable.

Clarifying whether such lattices exist would sharpen the relationship between shellability and CL-shellability in the specific context of lcm-lattices and could impact the understanding of the combinatorial-topological properties that correspond to algebraic properties of monomial ideals.