The multidimensional unique-active-transition idempotency conjecture

Let $d\in\mathbb{Z}_+$, and let $\tau:A^{\mathbb{Z}^d}\to A^{\mathbb{Z}^d}$ be a cellular automaton with a unique active transition $p\in A^S$, where $S\subset\mathbb{Z}^d$ is any finite subset such that $0\in S$. For $t\in S$, set

$$U:=S\cap(t+S).$$

Unique-active-transition idempotency conjecture. The automaton $\tau$ is not idempotent if and only if there exists $t\in S$ such that

$$p(t)\neq p(e)\quad\text{and}\quad p\vert_{U\setminus\{t\}}=p_{(-t+U)\setminus\{0\}}.$$

This conjecture would generalize the paper's characterization from one-dimensional cellular automata with interval neighborhoods to arbitrary finite neighborhoods and to cellular automata in every positive dimension. Its status is unresolved in the supplied source.