Uniqueness of the ${\mathrm{\Phi }}_3^4$ measures on closed Riemannian $3$-manifolds

We constructed in a previous work the ${\mathrm{\Phi }}_3^4$ measures on compact boundaryless $3$-dimensional Riemannian manifolds as some invariant probability measures of some Markovian dynamics. We prove in the present work that these dynamics have unique invariant probability measures. This is done by using an explicit coupling by change of measure that does not require any a priori information on the support of the law of the solution to the dynamics. In addition, the coupling can be used to see that the semigroup generated by the dynamics satisfies a Harnack-type inequality, which entails that the semigroup has the strong Feller property.