= Solution
Using $v=t+r_\star$ puts the <black string> metric into regular ingoing form,
$$
ds^2=-f\,dv^2+2\,dv\,dr+r^2d\Omega_3^2+dz^2.
$$
Its inverse metric gives
$$
\nabla^ar=(\partial_v)^a+f(\partial_r)^a,
\qquad
(\nabla r)^2=f.
$$
For $r<r_+$, $f<0$, so $\nabla r$ is timelike. It is future-directed by continuity from the future horizon, where it equals the future generator $\partial_v$. If $X$ is any future-directed causal tangent, then
$$
X(r)=g(X,\nabla r)<0.
$$
Thus $r$ decreases strictly along every future-directed causal curve in the interior. No such curve can cross back through $r=r_+$ or reach <future null infinity>. The region $r<r_+$ is consequently outside the causal past of future null infinity and is part of the <black hole> region.
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