Let $M$ be a smooth manifold and let $g(t)$, $t \in [0,T]$, be a smooth family of Riemannian metrics with nonnegative curvature operator. Suppose that, for each $t>0$, the null spaces of the curvature operator are invariant under parallel transport with respect to $g(t)$. If, for every $x \in M$ and every $t>0$, the holonomy representation on $\Lambda^2T_x^*M$ has no nonzero invariant subspace contained in the curvature-null space at $(x,t)$, then the curvature operator is positive for every point and every time $t>0$.