I'm learning about the (min-mean) cycle-cancelling alg for min-cost flow in Ahuja, Magnanti, and Orlan's Network Flows book (Chapters 9 and 10). When talking about the alg, they prove this fact related to the $\epsilon$-optimality of a flow:
Lemma 10.14: For a nonoptimal flow $x$, if we cancel a minimum mean cycle in $G(x)$, $\epsilon(x)$ cannot increase.
Does this still hold if we cancel a negative cycle, but not a min-mean negative cycle? We know that the $\epsilon(x) = \mu(x)$ - that is, $\epsilon(x)$ is equal to the mean weight of the min-mean cycle in $G(x)$. So, my question is equivalent to: if we cancel a negative cycle, can the mean weight of the min-mean negative cycle decrease?