Free Additive Convolution of Two Noncommutative Laws

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $\mathrm{add}=(A,0)$ be the affine datum from $2d$ to $d$ variables with $A_{ij}=1$ if $j=i$ or $j=d+i$, and $A_{ij}=0$ otherwise ($i\in[d]$, $j\in[2d]$); its tuple is $(x_{1}+x_{d+1},\dots,x_{d}+x_{2d})$.

¶For $\alpha,\beta\in\Sigma_{d}$, the free additive convolution of $\alpha$ and $\beta$ is the law

$$\alpha\boxplus\beta=(\alpha\star\beta)\circ\sigma_{\mathrm{add}}\in\Sigma_{d},$$

where $\alpha\star\beta\in\Sigma_{2d}$ is the free product (for $m=n=d$) and $\sigma_{\mathrm{add}}$ is the affine substitution; it belongs to $\Sigma_{d}$ by Affine Substitutions of Noncommutative Laws: Self-Adjointness, Composition, Moment Formulas and Positivity, and the Coordinate Data §self-adjoint.