Cyclic Symmetry of Nested Conditional Expectations over a Common Marginal

In the setting of Two Noncommutative Laws with a Common Marginal: Standing Notation for Their Amalgamated Free Product, with laws $\gamma_{1},\gamma_{2}$ of common marginal $\mu$, tracial algebras $N$ and $A_{\varepsilon}$ with traces $\tau_{\mu}$ and $\tau_{\gamma_{\varepsilon}}$, embeddings $\pi_{\varepsilon}$ and expectations $E_{\varepsilon}$, let $k\in\mathbb{N}$, let $f_{1},\dots,f_{k}\in\{1,2\}$, let $u_{j},v_{j}\in A_{f_{j}}$ for $j\in[k]$, and let $x,y\in N$. Define $G_{1},\dots,G_{k}\in N$ and $G_{1}',\dots,G_{k}'\in N$ recursively, as in Definition of Sequences by Recursion on the Natural Numbers §recursion, by

for $j\in[k]$ with $j\ge2$. ¶Then