The arbitrary-genus Lagrangian cobordism functoriality conjecture

Let $\mathcal{K}_0$ and $\mathcal{K}_1$ be Legendrian knots in $(Y,\xi)$, and suppose that $\mathcal{K}_0\prec_{\Sigma}\mathcal{K}_1$ is a Lagrangian cobordism of arbitrary genus. Write $K_0$ and $K_1$ for their underlying topological knot types, and let $\mathcal{R}$ be the coefficient ring. Lagrangian cobordism functoriality conjecture. There is a map

$$KHM(-Y,K_1)\otimes\mathcal{R}\longrightarrow KHM(-Y,K_0)\otimes\mathcal{R}$$

that sends $\ell(\mathcal{K}_1)$ to $\ell(\mathcal{K}_0)$. Such a map would extend the known functoriality of the invariant under the relevant Lagrangian cobordisms to cobordisms of arbitrary genus. The source gives no resolution of this conjecture.