Strict hierarchy of V-causality conditions for probabilistic cellular automata

Let $V^k$-causality denote the $k$-fold iterated screening-off-completability condition for stochastic maps over configurations, and let $V^kBox$ denote the corresponding family of stochastic maps. A stochastic map is $V^k$-causal when it satisfies the $k$-fold condition.

Strict V-causality hierarchy conjecture. For every $k$, there exists a $V^kBox$ which is $V^k$-causal but not $V^{k+1}$-causal.

This conjecture asserts that the hierarchy of increasingly strengthened causality conditions is strict: each level admits examples excluded by the next. The source presents the $k=1$ case as the previously described $VBox$ example; no resolution is supplied here.