Future causal geodesic completeness near expanding surface-symmetric cosmologies

Consider the setting of Theorem~

$$, for the surface-symmetric Einstein-Vlasov cosmological spacetimes discussed in the paper. **Future geodesic-completeness conjecture.** In the statement of Theorem~$$

, the spacetime is future causally geodesically complete. The surrounding discussion says that the answer is expected to be affirmative for $k\leq0$ and $\Lambda\geq0$, but the supplied text does not establish this conjecture in the stated generality.