A pair violating all three properties

Construct a pair of elements in a Bézout domain that violates good factorization, adequacy, and lcm splitting simultaneously.

Background

The paper gives examples separating the three properties but does not produce a single pair that fails all of them. It proposes using a Bézout domain associated with disjoint copies of the spectral construction as a possible route, while explicitly stating that the relevant details have not been checked.