Shereshevsky's finite-entropy conjecture for multidimensional cellular automata

Let $S$ be a finite set, let $d>1$, and let $X=S^{\mathbb{Z}^d}$. A $d$-dimensional cellular automaton is a continuous map $T:X\to X$ commuting with the shift action of $\mathbb{Z}^d$, and its topological entropy is denoted by $h_{\mathrm{top}}(T)$. Shereshevsky's conjecture. A $d$-dimensional cellular automaton with $d>1$ cannot have finite, positive topological entropy; equivalently, its entropy is either zero or infinite. The conjecture was motivated by the absence of forward-expansive cellular automata in dimensions greater than one and was known for linear cellular automata, but this paper disproves it by constructing, for every $d$, a $d$-dimensional cellular automaton with finite, nonzero topological entropy.