Uniqueness and equal distribution conjecture for least-gradient sections

Let $L$ be the flat affine bundle and let $v:M\to L$ be the BV section from Theorem~, obtained as a limit of the $q$-harmonic maps. Least-gradient section conjecture. The BV section $v:M\to L$ is unique, and the limit $v$ is equally distributed. Equivalently, the associated cohomology class and least-gradient section should be uniquely determined, with the resulting transverse measure giving equal weight to the components. The paper explains that uniqueness would determine the transverse measure associated with a possibly disconnected maximum stretch lamination, but the authors cannot prove it.