When are the marginals of an extremal invariant measure also extremal invariant?

Question

Let's suppose that $X$ is a compact metric space, and thus as is $X \times X$. If given a Markov process on $X \times X$ with marginals that are Markov processes on $X$, then we know that the marginals of any stationary/invariant measure are also invariant. Is the marginal of any probability measure that is extremal among the invariant measures for the 2-variate coupled process ("extremal" in the sense of convex analysis) also extremal among the invariant measures for the 1-variate process?