Convexity of the invariant coupling-operator set

Let $\mathcal{G}$ be the set of coupling operators and let $\mathcal{G}^0$ be the subset for which the corresponding Feller coupling has the required invariant measure. Convexity conjecture. The set $\mathcal{G}^0$ is convex. This is posed as an open question related to the convexity of $\mathcal{G}$, and the source gives no resolution.