Gluing Countably Many Square-Integrable Noncommutative Laws along a Common Marginal

In the setting of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation, let $k\in\mathbb{N}$. 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. $L^{2}$ tuples, their pairs and their laws are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples. Let $\pi\in\Sigma^{2}_{k}$, and for every $j\in\mathbb{N}$ let $m_{j}\in\mathbb{N}$ and $\gamma_{j}\in\Sigma^{2}_{k+m_{j}}$ satisfy $F^{m_{j}}_{\#}\gamma_{j}=\pi$.

¶There are a tracial W*-probability space $(H,M,\Omega)$, an $L^{2}$ $k$-tuple $Z$ of $(H,M,\Omega)$ and, for every $j\in\mathbb{N}$, an $L^{2}$ $m_{j}$-tuple $Y_{j}$ of $(H,M,\Omega)$, such that $\mathrm{law}(Z,Y_{j})=\gamma_{j}$ for every $j\in\mathbb{N}$.