Exhaustion of coupling operators by decomposed generator products

For $i \in \{1,\ldots,n\}$, let $(\mathcal{L}_{i},\mathcal{D}(\mathcal{L}_i))$ be generators of ergodic Feller semigroups. Let $A_k^i$, $B^i$, and $\mathcal{J}$ arise from decompositions of the marginal generators as in the source, and let $\bar{E}$ be the product state space. Exhaustion conjecture. There exist decompositions of the stated form and a set of coefficient functions

$$\mathcal{U} = \{(\alpha_{ijkl})_{(i,j,k,l)\in\mathcal{J}}:\bar{E}\rightarrow\mathbb{R}\}$$

such that

$$\mathcal{G}=\left\{\Gamma=\sum_{(i,j,k,l)\in\mathcal{J}}\alpha_{ijkl}A_k^iA_l^j:\alpha_{ijkl}\in\mathcal{U}\right\}.$$

The claim would show that the proposed construction exhausts all coupling operators. The source presents it as unresolved, and whether the constructed operators belong to $\mathcal{G}^0$ depends on the coefficients.