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on Strong block universality of Conway's Game of Life

The statement

Let $g:\{0,1\}^{\mathbb Z^2}\to\{0,1\}^{\mathbb Z^2}$ be Conway's Game of Life map. Does $g$ strongly simulate every block map $\phi:Y\to D^{\mathbb Z^2}$ whose domain $Y$ is a two-dimensional subshift of finite type? Explicitly, for every such $\phi$ , do there exist positive integers $m,n$ , an injective macrotile substitution $\tau:D\to\{0,1\}^{[m]\times[n]}$ , and a bijective partial block map $h$ from the Life configurations $x$ satisfying $g(x)\in\tau(D^{\mathbb Z^2})$ onto $Y$ , such that $g(x)=\tau(\phi(h(x)))$ for every such $x$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.