= Solution
Translation by $e=(0,1,0,\ldots,0)$ preserves $H$, so $(\omega(x+e):x\in H)$ has the same independent Bernoulli-$p$ law as $\omega$. This shift is ergodic, and the number $N$ of infinite clusters is shift-invariant. Thus $N=k_0$ almost surely for some deterministic $k_0\in\mathbb N\cup\{\infty\}$.
Suppose $2\leq k_0<\infty$. With positive probability a finite box meets all $k_0$ infinite clusters. By the <finite-energy property of Bernoulli percolation>, forcing finitely many sites in the box open has positive conditional probability and joins those clusters without affecting infinity outside the box. The resulting configuration has fewer than $k_0$ infinite clusters on an event of positive probability, contradicting the almost-sure constancy. Hence $k_0\in\{0,1,\infty\}$.
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