Maximality conjecture for the $coperatorname{L}^1$ metric on full groups of graphings

Let $\Phi$ be a graphing. The associated $\mathrm L^1$ full group consists of the measure-preserving transformations generated by the graphing, equipped with its natural $\mathrm L^1$ metric.

Maximality conjecture. The $\mathrm L^1$ metric on the $\mathrm L^1$ full group of $\Phi$ is maximal.

If true, this would imply that abstract isomorphisms between the relevant full groups are quasi-isometries and would remove the amenability and ergodicity hypotheses in the preceding theorem. The supplied text does not state whether the conjecture has been resolved.