Let $M$ be a closed smooth $n$-manifold, let $(g(t))_{t \in [0,T]}$ be a smooth Ricci flow on $M$, let $f: M \times [0,T] \to \mathbb{R}$ be smooth, and let $\tau: [0,T] \to (0,\infty)$ be smooth, all satisfying the hypotheses of Perelman's $\mathcal W$-entropy monotonicity theorem. If $J \subset [0,T]$ is a nondegenerate interval, then $\mathcal W[g(t),f(t),\tau(t)]$ is constant on $J$ if and only if

\begin{align*} \operatorname{Ric}_{g(t)} + \operatorname{Hess}_{g(t)} f(t) = \frac{1}{2\tau(t)}g(t) \end{align*}

at every point of $M$ and for every $t \in J$.