Convexity of the coupling-operator set

Let $(\bar{E},\bar{\tau})$ be the product state space, let $(\mathcal{L}_i,\mathcal{D}(\mathcal{L}_i))$ be the generators of the marginal Feller semigroups, and let $\mathcal{G}$ denote the set of coupling operators satisfying the coupling-operator conditions. Convexity conjecture. The set $\mathcal{G}$ is convex. This would provide a useful structural property for constructing and comparing couplings. The source does not provide a resolution.