By work of Berglund and Madsen, the rings of rational characteristic classes of fibrations and smooth block bundles with fibre $D2n♯(Sn\times Sn)♯g$, relative to the boundary, are for $2n\ge 6$ independent of $g$ in degrees $\ast \le (g-6)/2$. In this note, we explain how this range can be improved to $\ast \le g-2$ using cohomological vanishing results due to Borel and classical invariant theory. This implies that the analogous ring for smooth bundles is independent of $g$ in the same range, provided the degree is small compared to the dimension.