Extension of acausal surfaces to Cauchy surfaces for cone-bundle embeddings

Let $C^\first$ and $C^{\first}$ be globally hyperbolic spacelike cone bundles over $M$ and $M'$, respectively, and let $\chi\colon M\to M'$ be an embedding inducing a chronologically compatible morphism $T^*\chi\colon C^\first\to C^{\first}$. If $\Sigma\subset M$ is a $C^\first$-Cauchy surface and $K\subseteq\Sigma$ is compact, then there is a neighborhood $U$ of $K$ in $\Sigma$ and a $C^{\first}$-Cauchy surface $\Sigma'\subset M'$ such that $\Sigma'$ agrees with $\chi(\Sigma)$ on $\chi(U)$. Cauchy-surface extension conjecture. Under these hypotheses, such a surface $\Sigma'$ exists for every compact $K\subseteq\Sigma$. This would support the construction of locally covariant field theories by providing the Cauchy-surface extension needed for the Isotony property; its validity is left for future work.