  This note is a continuation of [CMZ21]. We shall show that an ancient Ricci
flow with uniformly bounded Nash entropy must also have uniformly bounded
$\nu$-functional. Consequently, on such an ancient solution there are uniform
logarithmic Sobolev and Sobolev inequalities. We emphasize that the main
theorem in this paper is true so long as the theory in [Bam20c] is valid, and
in particular, when the underlying manifold is closed.
