Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal

In the setting of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation, let $k,m,n\in\mathbb{N}$. For $l\in\mathbb{N}$ let $F^{l}=(E^{l},0)$ be the affine datum from $k+l$ to $k$ variables with $E^{l}_{ij}=1$ if $j=i$ and $E^{l}_{ij}=0$ if $j\ne i$ ($i\in[k]$, $j\in[k+l]$), so that $F^{l}_{\#}$ is the push-forward to the law of the first $k$ variables. Let $\pi_{1}\in\Sigma^{2}_{k+m}$ and $\pi_{2}\in\Sigma^{2}_{k+n}$ satisfy $F^{m}_{\#}\pi_{1}=F^{n}_{\#}\pi_{2}$.

¶There are a tracial W*-probability space $(H,M,\Omega)$, an $L^{2}$ $k$-tuple $X$, an $L^{2}$ $m$-tuple $Y$ and an $L^{2}$ $n$-tuple $Z$ of $(H,M,\Omega)$ with $\mathrm{law}(X,Y)=\pi_{1}$ and $\mathrm{law}(X,Z)=\pi_{2}$.