= Solution
\b[For the printed <spacetime>, yes.] The surfaces $T=\text{constant}$, equivalently $t=\text{constant}$, are <Cauchy surfaces> in <Minkowski spacetime>. Every inextendible causal curve intersects each once; if its monotone time coordinate had a finite endpoint, the bound $|dx/dT|\leq1$ would give a finite spatial endpoint too, allowing continuation. Causal diamonds are closed and bounded in $(T,x)$ and hence compact. Therefore the printed <spacetime> is <globally hyperbolic>.
Initial data on a Cauchy slice determine unique causal evolution for the usual well-posed hyperbolic field equations, without boundary data at spatial infinity at finite time. The time reparametrization preserves that property.
\b[For the qualified AdS interpretation, no.] On the <universal cover>, choose two central events with enough time separation for a null ray to approach a strip edge and return. Their causal diamond contains a sequence with $\rho\to\pi/2$ but no limit in the physical <spacetime>, so it is noncompact. A constant-$t$ surface fails to meet every inextendible causal curve arriving from timelike conformal infinity. Thus even the cover is not <globally hyperbolic>. Field evolution needs appropriate boundary conditions, such as a reflecting condition compatible with the field theory, as well as initial data. If AdS time itself is periodically identified, <closed timelike curves> give an additional and more immediate obstruction. These are consequences of the alternative spatial-lapse geometry, not of the printed flat <metric tensor>.
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