Cameron–Glynn algebra isomorphism conjecture for age-inexhaustible structures

Let $\mathcal{M}$ be an age-inexhaustible relational structure, meaning that any two finite substructures of $\mathcal{M}$ have disjoint copies within $\mathcal{M}$. Let the Cameron and Glynn algebras be the two algebras associated with the automorphism group of $\mathcal{M}$. Cameron–Glynn isomorphism conjecture. The Cameron and Glynn algebras are isomorphic. This conjecture proposes an algebraic coincidence for age-inexhaustible relational structures; the surrounding discussion relates age-inexhaustibility to integral-domain properties, but gives no resolution of the isomorphism claim.