Let $I(X,Y):=H(X)+H(Y)-H(X,Y)$ be the mutual information of the joint probability distribution $p_{XY}$ (here $H(\cdot)$ is the Shannon entropy of its argument). I know that the mutual information is invariant IF $X'=\text{invertible_function}(X)$ and $Y'=\text{invertible_function}(Y)$, that is, it is invertible under a re-parametrization.

Is the converse true? I.e., is it true that $$I(X,Y) = I(X', Y')$$ implies that the marginals $X', Y'$ are invertible functions of $X$ and $Y$, respectively?

Thanks!