Let $k$ be a field, and let $q\in k^{\times}$. We can then consider the Temperley-Lieb category $TL(q)$. The objects of $TL(q)$ are the non-negative integers, and morphisms are roughly isotopy classes of string diagrams (without crossings) contained in a planar strip up to the relation that any circle evalutes to $-(q^{-1}+q)$. This category is monoidal, and objects have duals. In fact, it has a canonical spherical structure.

It is quite standard that, provided that $q$ admits a square root $q^{1/2}$ in $k$, then $TL(q)$ can be endowed with a braiding. Explicitly, it is given by Kauffman's skein relation. Now, I am fairly sure that the answer to the following question is yes, but I haven't been able to find a reference in the literature.

Are these braidings unique? More precisely, is it true that (the equivalence classes of) braidings on $TL(q)$ are classified by the square roots of $q$ in $k$?

This question was asked on MS first, but wasn't answered there.