Nontermination conjecture for the supercritical fully packed $O(2)$ iteration

Fix $s>1$. In the modified iteration procedure, each hole of perimeter $m$ is filled using a ring whose inner boundary has length either $\lfloor sm\rfloor$ or $\lfloor s^{-1}m\rfloor$, with the two choices made equiprobably and independently. Let $k$ be the initial boundary length. Supercritical iteration nontermination conjecture. With probability tending to one as $k\to\infty$, the procedure does not terminate after finitely many steps. Consequently, with high probability for large $k$, iterating countably many times yields an infinite triangulation $M_k'$ with boundary perimeter $k$, decorated by an infinite fully packed loop configuration $L_k'$ on its dual. The conjecture is motivated by the expected infinite graph-distance diameter of the corresponding maps and the expectation that the associated multi-type branching process is supercritical; it remains open.