= Solution
First, causal futures of black-hole points remain in the black-hole region. Indeed, if $p\in B$ and $q\in J^+(p)$ could send a signal to <future null infinity>, concatenating the causal curves would put $p$ in $J^-(\mathcal I^+)$, a contradiction. Hence
$$
J^+(B)\cap\Sigma_2\subset B\cap\Sigma_2.
$$
Global hyperbolicity supplies the following connectedness lemma: if $C$ is connected on one <Cauchy hypersurface>, then $J^+(C)$ intersects any later Cauchy hypersurface in a connected set. To see the relevant mechanism, flow $C$ to the later surface along a continuous future timelike vector field; the image is connected, and every additional causally reachable point is joined to that image by the endpoint deformation of a causal curve inside the globally hyperbolic diamond. Applying the lemma to the connected component $B$ shows that $J^+(B)\cap\Sigma_2$ is connected. Since it is contained in $B\cap\Sigma_2$, it must lie wholly inside one connected component of that set. This is the <black-hole non-splitting theorem>: later black-hole components may merge, but one earlier connected black hole cannot split into two future components.
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