Galton–Watson distribution conjecture for the infinite Schaeffer tree

Let ${\rm Sch}_\infty$ be the infinite Schaeffer tree, and let $\Delta h$ be the antisymmetric function on its directed edges defined by the increments of the Busemann function. Consider the pair $({\rm Sch}_\infty,\Delta h)$.

Distribution conjecture. The pair $({\rm Sch}_\infty,\Delta h)$ is distributed as a Galton–Watson tree with geometric-$1/2$ offspring distribution, conditioned to be infinite, with increments distributed independently and uniformly among $\{-1,0,1\}$.

The conjecture seeks an explicit probabilistic description of the infinite Schaeffer tree and its edge increments. The supplied text gives no resolution, so it remains open.