Extension of the uniqueness theorem beyond ${\mathcal A}$

Let ${\mathcal A}$ be the class of probability measures appearing in Theorem, let $\mu$ be an invariant probability measure, and let the theorem's part b) assert uniqueness under the assumption $\mu\in{\mathcal A}$. Extension of the uniqueness theorem. Part b) of Theorem continues to hold true without the assumption $\mu\in {\mathcal A}$. The conjecture proposes that the uniqueness conclusion does not require membership in ${\mathcal A}$; the supplied context does not state whether this has been resolved.