The conjecture on regular equivalence classes of Haagerup monsters

A Haagerup monster is a group constructed by the method of the paper that has the Haagerup property and contains the prescribed obstruction to coarse embeddability. Two such groups are regularly equivalent when there are regular maps in both directions, where a map between graphs is regular if it is Lipschitz and inverse images of vertices have uniformly bounded cardinality. The notation $2^{\aleph_0}$ denotes the cardinality of the continuum.

Regular equivalence-class conjecture. There exists $2^{\aleph_0}$ regular equivalence classes of Haagerup monsters.

The conjecture asks for continuum many mutually distinct regular-equivalence classes among the counterexamples produced by the paper, strengthening the quantitative separation phenomena described for expanders, graphs with walls, and special Gromov monsters. The source gives no resolution.