In this paper we continue to explore infinitely renormalizable H\'enon maps
with small Jacobian. It was shown in [CLM] that contrary to the one-dimensional
intuition, the Cantor attractor of such a map is non-rigid and the conjugacy
with the one-dimensional Cantor attractor is at most 1/2-H\"older. Another
formulation of this phenomenon is that the scaling structure of the H\'enon
Cantor attractor differs from its one-dimensional counterpart. However, in this
paper we prove that the weight assigned by the canonical invariant measure to
these bad spots tends to zero on microscopic scales.