Freeness of Two Groups of Variables under a Noncommutative Law

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $m,n\in\mathbb{N}$. In this item $\iota^{1}:\mathcal{P}_{m}\to\mathcal{P}_{m+n}$ and $\iota^{2}:\mathcal{P}_{n}\to\mathcal{P}_{m+n}$ are the substitutions of the tuples $(x_{1},\dots,x_{m})$ and $(x_{m+1},\dots,x_{m+n})$ of $\mathcal{P}_{m+n}$; for $m=n=d$ they are the marginal substitutions. For $\gamma\in\Sigma_{m+n}$, the marginals $\gamma\circ\iota^{1}$ and $\gamma\circ\iota^{2}$ belong to $\Sigma_{m}$ and $\Sigma_{n}$ by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §law, the variables lying in $\mathcal{P}_{m+n,\mathrm{sa}}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint.

¶A law $\gamma\in\Sigma_{m+n}$ is free (the first $m$ and the last $n$ variables are free under $\gamma$) if

$$\gamma\bigl(\iota^{e_{1}}(p_{1})\,\iota^{e_{2}}(p_{2})\cdots\iota^{e_{k}}(p_{k})\bigr)=0$$

for every $k\in\mathbb{N}$, every $k$-tuple $(e_{1},\dots,e_{k})$ in $\{1,2\}$ with $e_{i}\neq e_{i+1}$ whenever $i,i+1\in[k]$, and all polynomials $p_{1},\dots,p_{k}$ with $p_{i}\in\mathcal{P}_{m}$ if $e_{i}=1$, $p_{i}\in\mathcal{P}_{n}$ if $e_{i}=2$, and $\gamma(\iota^{e_{i}}(p_{i}))=0$ for every $i\in[k]$.