Ah, I guess @NoamD.Elkies's comment about no configuration being known that can survive an arbitrary finite barrage of gliders answers my question.

Is the variant of Conway's question in which $c$ is only required to be live infinitely often open? Or with 'periodically'?

@TimothyChow In my field, Seva's usage of $o(1)$ is perfectly standard. The statement that "$|A| > (1+o(1))\sqrt{2p}$ suffices" means: there is some function $r$ with $r = o(1)$ such that $|A| > (1+r(p))\sqrt{2p}$ suffices.