Salo–Törmä question on the Game of Life limit set

Let $F:\{0,1\}^{\mathbb{Z}^2}\to\{0,1\}^{\mathbb{Z}^2}$ be the global map of Conway's Game of Life, and let $X=\{x\in\{0,1\}^{\mathbb{Z}^2}:\forall n\ge 0\ \exists y\in\{0,1\}^{\mathbb{Z}^2}\text{ such that }F^n(y)=x\}$ be its limit set. Is the translation action of $\mathbb{Z}^2$ on $X$ topologically transitive? Equivalently, for every pair of finite patterns $p$ and $q$ that each occur in some configuration of $X$, does there exist a configuration $x\in X$ and a translation vector $v\in\mathbb{Z}^2$ such that $x$ contains $p$ and also contains $q$ translated by $v$?