  In (Calc.Var.PDE 2018) Schultz generalized the work of Rajala and Sturm
(Calc.Var.PDE 2014), proving that a weak non-branching condition holds in the
more general setting of very strict CD spaces. Anyway, similar to what happens
for the strong CD condition, the very strict CD condition seems not to be
stable with respect to the measured Gromov Hausdorff convergence. In this
article I prove a stability result for the very strict CD condition, assuming
some metric requirements on the converging sequence and on the limit space. The
proof relies on the notions of \textit{consistent geodesic flow} and
\textit{consistent plan selection}, which allow to treat separately the static
and the dynamic part of a Wasserstein geodesic. As an application, I prove that
the metric measure space $\mathbb R^N$ equipped with a crystalline norm and
with the Lebesgue measure satisfies the very strict $\ CD(0,\infty)$ condition.
