  We study the problem of determining if the braid group representations
obtained from quantum groups of types $E, F$ and $G$ at roots of unity have
infinite image or not. In particular we show that when the fusion categories
associated with these quantum groups are not weakly integral, the braid group
images are infinite. This provides further evidence for a recent conjecture
that weak integrality is necessary and sufficient for the braid group
representations associated with any braided fusion category to have finite
image.
