On connection between associators and 6j-symbols

We propose a new formula relating the rationalized Drinfeld associators obtained from the Knizhnik–Zamolodchikov equation to the quantum $6j$-symbols of the algebra ${\mathcal{U}}_q(\mathfrak{g})$: $U_q=U\cdot {\mathrm{\Phi }}_{\mathbb{Q}}^{KZ}$. We prove this relation in the case of the ${\mathfrak{s}\mathfrak{l}}_2$ Lie algebra for the tensor cube of the fundamental and the first symmetric representation. We also check it perturbatively for the ${\mathfrak{s}\mathfrak{l}}_N$ algebra for representations $[2]\otimes [1]\otimes [1]$, $[2]\otimes [2]\otimes [1]$, the non-trivial cases with multiplicities — $[2,1{]}^{\otimes 3}$, $[2,2{]}^{\otimes 3}$, and in the case of the ${\mathfrak{s}\mathfrak{o}}_5$ Lie algebra for the tensor cube of the fundamental representation.