Asymptotic disconnectedness 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 connected when it induces a connected subgraph of the permutohedron, with vertices the linear orders and edges joining orders differing by one inversion. A property holds asymptotically almost surely when its proportion tends to $1$ as $n$ tends to infinity.

Disconnectedness conjecture. Asymptotically almost surely, MUCDs are not connected.

The claim is based on the enumeration through degree $7$, where most MUCDs are not connected, but no proof or resolution is supplied here.