The short answer to your question is that if $x,y$ are elements in an algebra in topologically free $k[[\hbar]]$-modules whose constant term is 1, then they have a unique square root whose constant term is also one, and if $x,y$ commute then say the square root of $x$ also commutes with $y$. Indeed if $a$ is the square root of $x$, then
$$(yay^{-1})^2=ya^2y^{-1}=x$$
and because $yay^{-1}$ also has constant term equal to 1, we get $yay^{-1}=a$. This shows at once that $(RR^{2,1})^{\frac12}$ commutes with $R$.

One way to think about it is as follow (this is also explained in Joel's paper). Any finitely generated group $G$ has a so-called pro-unipotent aka malcev aka rational completion $G(\mathbb{Q})$. One of its definition is that it is the univrsal uniquely divisible group having a morphism from $G$. In other words, it is the universal group in which images of elements of $G$ have a unique $n$th root for any $n$. So roughly elements of this groups are the $x^{\lambda}$ where $x \in G$ and $\lambda \in \mathbb{Q}$. Now the same argument as above shows if $x,y$ commute, then so do any possibly rational power of their image in $G(\mathbb{Q})$ (uniqueness is again key here).

This has a relative version, where in the case at hand you roughly speaking apply this construction to the pure braid group $P_n$ inside of the braid group $B_n$: you get a certain group $B_n(\mathbb{Q})^{rel}$ fitting into an exact sequence
$$1 \rightarrow P_n(\mathbb{Q}) \rightarrow B_n(\mathbb{Q})^{rel} \rightarrow S_n\rightarrow 1. $$

Long story short you get this way a morphism from the so-called cactus group $\Gamma_n$ (the group of which coboundary categories give representations) into $B_n(\mathbb{Q})^{rel}$ by taking square roots of the generators of $P_n$ inside there. Now for any quantized quasi-Hopf algebra, of more generally in any braided tensor category over $k[[\hbar]]$ in which the braiding satisfies
$$\beta_{U,V}\beta_{V,U} =id_{U\otimes V} +O(\hbar)$$

the representations of $B_n$ you get factor through $B_n(\mathbb{Q})^{rel}$, hence restrict to representations of $\Gamma_n$.

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