Gluing Two Noncommutative Couplings along a Common Marginal

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $d\in\mathbb{N}$, let $R>0$ be real, let $\mu,\nu,\rho\in\Sigma_{d,R}$, and let $\gamma_{1}\in\Pi(\mu,\nu)$ and $\gamma_{2}\in\Pi(\nu,\rho)$ be couplings. Let $\sigma^{12}$, $\sigma^{23}$ and $\sigma^{13}$ be the substitutions, from $\mathcal{P}_{2d}$ to $\mathcal{P}_{3d}$, of the $2d$-tuples

in $\mathcal{P}_{3d}$. Then there is $\psi\in\Sigma_{3d,R}$ with the following properties.

1. (First pair)¶ $\psi\circ\sigma^{12}=\gamma_{1}$.

2. (Second pair)¶ $\psi\circ\sigma^{23}=\gamma_{2}$.

3. (Composite coupling)¶ $\psi\circ\sigma^{13}\in\Pi(\mu,\rho)$.