Converse of Hamilton's Maximum Principle?

The famous maximum principle of Hamilton states the following. Let $C$ be a convex $O(n)$-invariant subset of the space of algebraic curvature operators. Then if it is invariant under the ODE $$\dot{R} = R^2 + R^\#,$$ (where $R^\#$ is quadratic in $R$ and is defined in terms of the adjoint action), then it is invariant unter the Ricci flow, i.e. a Ricci flow whose curvature starts in $C$ will remain in $C$ for all times where it is defined.

What is known about the converse? If a set $C$ is invariant under Ricci flow, must $C$ be invariant under the ODE? If not, are there counter-examples?