Proposition: The regular product is not globally hyperbolic

The manifold $M$ has a noncompact causal diamond.

Proof. Fix $r_0>0$, an angle, and $z=0$. Take endpoints at that spatial position and times $T_{\ast }\pm 2r_0/c$. For $r_n\downarrow 0$, the events at time $T_{\ast }$, radius $r_n$, and the same angle and axial coordinate lie in the diamond, using radial causal paths. They have no convergent subsequence in $M$.