The asymptotic number of probabilistic compositional graphoids

A compositional graphoid is a conditional-independence structure satisfying the Composition axiom, and a compositional graphoid is representable by $n$ discrete random variables if it is induced by the conditional-independence relations among those variables. The notation $\Omega(n^\varepsilon)$ denotes an asymptotic lower bound up to a positive constant factor.

Asymptotic counting conjecture. The number of compositional graphoids which are representable by $n$ discrete random variables is asymptotically

$$2^{2^{\Omega(n^\varepsilon)}}$$

for some $\varepsilon>0$ (or even $\varepsilon>1$).

This conjecture asks whether probabilistic compositional graphoids are doubly exponentially numerous, paralleling known lower bounds for probabilistic conditional-independence structures and the scarcity of graphical representations. The source proposes the asymptotic estimate but gives no resolution.