Let $(M,g(\tau))$ be a complete backward Ricci flow on $\tau\in(0,t_0]$ with bounded curvature on compact subintervals of $(0,t_0]$, based at $(p,0)$. Let $l:M\times(0,t_0]\to\mathbb R\cup\{\infty\}$ be the reduced distance, and suppose $(q,\tau)$ is reached by a minimizing $\mathcal L$-geodesic and is a smooth point of $l$. At such a point,

\begin{align*} \partial_\tau l-\Delta l+|\nabla l|^2-R+\frac{n}{2\tau}\ge 0. \end{align*}

On $M\times(0,t_0]$, the same inequality holds in the weak upper-barrier sense across the $\mathcal L$-cut locus wherever $l$ is finite.