Compactified Gaussian free field conjecture for Temperleyan forest height forms

Let $M$ be a Riemann surface, and consider the scaling limit of Temperleyan forests on $M$ together with their height forms. In the general case, where the Euler characteristic $\chi$ need not vanish, the limiting height form is conjectured to be the compactified Gaussian Free Field on the appropriately punctured surface.

Compactified Gaussian free field conjecture. The limiting height form is given by the compactified Gaussian Free Field in the appropriately punctured surface.

The scaling limit is proved in the case $\chi=0$, while extending the scaling-limit result to general topology requires proving existence of the scaling limit for the special branches. Identification of the general limiting law remains open; in particular, it is unknown whether it is given by the compactified Gaussian Free Field with appropriate punctures.