The KZ equations on the configuration space of $n$ distinct points in $\mathbb C$ give rise to a representation of $B_n$ on $V^{\otimes n}$, where $V$ is any given representation of $SL(2)$ (we'll stick to this case; clearly we could work with other Lie groups as well). The WRT Hilbert space $\mathcal H_n$ associated to the $SL(2)$ character variety of $\mathbb S^2\setminus\{p_0,\ldots,p_n\}$ (where we restrict the monodromy around each $p_i$ to lie in a particular conjugacy class) also carries a natural action of $B_n$.

Now, both of these representations give rise to the Jones polynomial in a more or less straightforward manner, so I would almost like to conjecture that there is a natural isomorphism between the two. Unfortunately, I don't even see a reason why they should have the same dimension (and I'd guess they probably don't!). Thus I have a weaker proposal:

Question: Is there a natural homomorphism $\rho:V^{\otimes n}\to\mathcal H_n$ (or perhaps it should go the other direction) which respects the action of $B_n$?

Remark: There are choices to be made to construct these representations (KZ and WRT). I expect the choice of representation $V$ and Planck constant $\hbar$ in the KZ equations correspond in a straightforward manner with the choice of level $k$ and conjugacy class for the WRT character variety. More specifically, one would most certainly take $\hbar=k^{-1}$ and the conjugacy class to have trace $2\cos(\hbar(\dim V-1))$ [perhaps up to multiplicative constants].