Least-gradient primitive conjecture for measured laminations

Let $M$ be the surface under consideration and let $\lambda$ be an arbitrary oriented lamination on $M$ with a transverse measure. Let $\tilde v:\tilde M\to\mathbb R$ be a primitive of the Ruelle--Sullivan current associated with the transverse measure. For any ball $B\subset\tilde M$, restrict $\tilde v$ to $B$. Least-gradient primitive conjecture. The restriction $\tilde v\big|_B$ is of least gradient. The proposed partial converse is motivated by the expectation that the boundaries of the superlevel sets $\{\tilde v\ge t\}$ in $B$ are geodesics, which should imply local least-gradient behavior.