Monotone first-marginal conjecture for separable supermodular VMOT

Let $d \ge 2$, let $\mu_i \preceq_c \nu_i$ for $i=1,\ldots,d$, and let $\vec\mu=(\mu_1,\ldots,\mu_d)$. A monotone coupling of $\vec\mu$ is the unique monotone probability measure with marginals $\mu_1,\ldots,\mu_d$, denoted by $\chi_{\vec\mu}$. Let the cost be

$$c(x,y)=c_1(x)+c_2(y),$$

for $x,y\in\mathbb{R}^d$, where $c_1$ and $c_2$ are supermodular, and let a VMOT denote an optimizer for the problem referred to as VMOT. Monotone first-marginal conjecture. There exists a VMOT $\pi$ whose first-time marginal $\pi^X$ is the monotone coupling $\chi_{\vec\mu}$. Moreover, if $c_1$ is strictly supermodular, then every VMOT $\pi$ satisfies

$$\pi^X=\chi_{\vec\mu}.$$

The claim would identify the optimal first-time dependence structure in this separable-cost martingale transport problem; the supplied text does not state whether it has been proved or remains open.