The ultra-fat half-grid conjecture for quasi-transitive locally finite graphs

Let $G$ be a connected, quasi-transitive, locally finite graph. An end of $G$ is thick if it contains infinitely many pairwise disjoint rays, and an ultra-fat model of the half-grid is a model of the half-grid in $G$ satisfying the ultra-fatness condition defined in the paper.

Ultra-fat half-grid conjecture. Every connected, quasi-transitive, locally finite graph with a thick end contains an ultra-fat model of the half-grid.

This assertion would suffice to extend the equivalence between the existence of a thick end, the presence of the half-grid as an ultra-fat, asymptotic, or diverging minor, and failure to be quasi-isometric to a tree from accessible graphs to all connected, quasi-transitive, locally finite graphs. Its status is not resolved in the supplied text.