unique optimal plan in $\Gamma_o(\gamma,\nu)$?

Question

Let $\gamma\in \mathcal{P}(X^2)$ (set of probability measures on $X^2$) and $\mu:=\pi^1_{\#}\gamma$ (first marginal of $\gamma$). Define $$\Gamma_o(\gamma,\nu):=\left\{\eta\in\mathcal{P}(X^3):(\pi^1,\pi^2)_{\#}\eta=\gamma,(\pi^1,\pi^3)_{\#}\eta\in\Gamma_o(\mu,\nu)\right\}.$$ I want to understand why if $\gamma=(i\times\xi)_{\#}\mu$ is induced by a transport map and $\mu$ is absolutely continuous, then $\Gamma_o(\gamma,\nu)$ contains only one element $$\Gamma_o(\gamma,\nu)=\{(i\times\xi\times t_{\mu}^{\nu})_{\#}\mu\}$$