Asymptotic irreducibility conjecture for maximal uniquely maximal Condorcet domains

Let a maximal uniquely maximal Condorcet domain (MUCD) be a maximal Condorcet domain of the type considered in the paper, and call it irreducible when no proper subset of at least two alternatives is consecutive in every ranking in the domain. A property holds asymptotically almost surely when the proportion of MUCDs having that property tends to $1$ as $n$ tends to infinity.

Irreducibility conjecture. MUCDs are asymptotically almost surely irreducible.

The claim is motivated by the enumeration through degree $7$, where reducible MUCDs become more numerous but are expected to be asymptotically outnumbered by irreducible ones. No resolution is given in the supplied text.