  Consider a Hamiltonian circle action on a closed $8$-dimensional symplectic
manifold $M$ with exactly five fixed points, which is the smallest possible
fixed set. In their paper, L. Godinho and S. Sabatini show that if $M$
satisfies an extra "positivity condition" then the isotropy weights at the
fixed points of $M$ agree with those of some linear action on $\mathbb{CP}^4$.
Therefore, the (equivariant) cohomology rings and the (equivariant) Chern
classes of $M$ and $\mathbb{CP}^4$ agree; in particular, $H^*(M;\mathbb{Z})
\simeq \mathbb{Z}[y]/y^5$ and $c(TM) = (1+y)^5$. In this paper, we prove that
this positivity condition always holds for these manifolds. This completes the
proof of the "symplectic Petrie conjecture" for Hamiltonian circle actions on
on 8-dimensional closed symplectic manifolds with minimal fixed sets.
