  Loeper's condition in \cite{Loe09} and the quantitatively quasi-convex
condition (QQconv) from \cite{GK15} are synthetic expressions of the analytic
MTW condition from \cite{TW} since they only require $C^2$ differentiability of
the cost function $c$. When the cost function $c$ is $C^4$, it is known that
the two synthetic MTW conditions are equivalent to the analytic MTW condition.
However, when the cost function has regularity weaker than $C^4$, it is not
known that if the two synthetic MTW conditions are equivalent. In this paper,
we show the equivalence of the synthetic MTW conditions when the cost function
has regularity slightly weaker than $C^3$.
