Let $X,Y$ be two measurable spaces, $\mu,\nu$ two probability measures on $X$, and $\kappa$ a transition kernel from $X$ to $Y$. Define $\tilde\mu(dy)=\int_X\kappa(dy|x)\mu(dx)$ and $\tilde\nu(dy)=\int_X\kappa(dy|x)\nu(dx)$.

Denote by $\Gamma(\mu,\nu)$ the set of couplings of $\mu$ and $\nu$ and by $\Gamma(\tilde\mu,\tilde\nu)$ the corresponding set for $\tilde\mu$ and $\tilde\nu$.

Furthermore, let $S_{\kappa}:\Gamma(\mu,\nu)\to\Gamma(\tilde\mu,\tilde\nu)$ be the mapping defined by

$$S_{\kappa}(\pi)(dy_1,dy_2)=\int_{X\times X}\kappa(dy_1|x_1)\kappa(dy_2|x_2)\pi(dx_1,dx_2),\quad \pi\in\Gamma(\mu,\nu).$$

Question: Is $S_{\kappa}$ onto $\Gamma(\tilde\mu,\tilde\nu)$?