It is definitely not known if $\zeta(3)/\pi^3$ is rational or not. By the way, there is a paper of Felder and Willwacher where they prove that the weight of a certain graph appearing in Kontsevich's formality quasi-isomorphism is, up to a rational, $\zeta(3)/\pi^3$. The question whether Kontsevich's quasi-isomorphism is defined of $\mathbb{Q}$ or not, is still open. If the answer to this question would be "yes", then the associator defined by Alekseev and Torossian would have rational coefficients... and that would definitely be a great result!
DamienC
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