MC-finiteness of labeled ternary spaces

Determine the number of labeled ternary spaces on an n-element universe satisfying axioms $T_1$–$T_4$, and establish whether the resulting counting sequence is MC-finite.

Background

Ternary spaces are introduced as ternary structures satisfying four axioms governing a betweenness-like relation. They form a natural class of finite ternary structures, but their density function is not covered by the paper’s bounded-G-degree theorem in general. The authors explicitly leave both the enumeration of labeled ternary spaces and the MC-finiteness of that enumeration unresolved.