Asymptotic non-normality 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. A Condorcet domain is normal when it is isomorphic to one containing both the standard order and its reverse. A property holds asymptotically almost surely when its proportion tends to $1$ as $n$ tends to infinity.

Non-normality conjecture. Asymptotically almost surely, MUCDs are not normal.

The claim is motivated by the enumeration, which shows substantially fewer normal MUCDs than total MUCDs; no proof or resolution is given in the supplied text.