Joint interface-function convergence for bipolar-oriented planar maps

Consider the NW/SE and NE/SW interface functions associated with a random bipolar-oriented planar map. In the scaling limit, the NW/SE interface function is a Brownian excursion, and the corresponding continuum object is an $\operatorname{SLE}_{12}$-decorated $\sqrt{4/3}$-LQG surface.

Joint interface-function convergence conjecture. The joint law of both NW/SE and NE/SW interface functions of a random bipolar-oriented planar map converges to the joint law of both corresponding interface functions of $\operatorname{SLE}_{12}$-decorated $\sqrt{4/3}$-LQG.

The conjecture addresses the joint convergence of the two tree-interface descriptions. The source explains that, although the discrete NW/SE interface functions determine the map and the continuum NW/SE Brownian excursion determines the SLE-decorated LQG, convergence of the jointly specified pair is the issue under consideration.