Representative-independence conjecture for the causal hypersurface condition

Let $(\mathcal M,g)$ be a generalized space-time with a symmetric representative $(g_\varepsilon)_\varepsilon$, and let $\Sigma$ be a past-compact space-like hypersurface. For sufficiently small $\varepsilon$, write $J_\varepsilon^+(\Sigma)$ for the topological closure in the relevant open set of the future emission of $\Sigma$ with respect to $g_\varepsilon$. Representative-independence conjecture. If, for one symmetric representative, $\Sigma$ satisfies

$$\partial J_\varepsilon^+(\Sigma)=\Sigma$$

for sufficiently small $\varepsilon$, then it satisfies the corresponding condition for every symmetric representative of $g$.

The condition is used to ensure existence of smooth representative-level solutions to the wave equation. The source proposes invariance under changing the symmetric representative but gives no resolution.