The N-regularity conjecture for mutable lattices

Let $L$ be a mutable lattice, and let $\operatorname{Hasse}(L)$ denote its Hasse diagram. Consider the underlying undirected graph of this diagram, and let $N$ be a constant.

N-regularity conjecture. Every vertex of the underlying undirected graph of $\operatorname{Hasse}(L)$ has degree $N$; equivalently, the graph is $N$-regular.

The paper states that the converse is false, so $N$-regularity does not characterize mutable lattices. The conjecture itself is not resolved in the supplied text.