The converse of a theorem by Bayer and Stillman

Bayer-Stillman showed that $reg(I) = reg(gin_\tau(I))$ when $\tau$ is the graded reverse lexicographic order. We show that the reverse lexicographic order is the unique monomial order $\tau$ satisfying $reg(I) = reg(gin_\tau(I))$ for all ideals $I$. We also show that if $gin_{\tau_1}(I) = gin_{\tau_2}(I)$ for all $I$, then $\tau_1 = \tau_2$.