Markov field property for local causal states

Let the local causal state at a space-time point be denoted by $S(\left\langle v,t\right\rangle)$. The parents of this state are the causal states one time-step before $\left\langle v,t\right\rangle$ and inside its past light cone, namely

$$A(\left\langle v,t\right\rangle) = \left\{ S(\left\langle u,t-1\right\rangle) \mid (u=v) \vee (uEv) \right\}.$$

Markov field property. The local causal states form a Markov field in space-time.

This would strengthen the temporal Markov property, which states that a local causal state is independent of the local causal states in its past light cone, given its parents. The source presents this stronger spatial-temporal Markov property as desirable, but provides no resolution here.