The Alcalde Lopez-Heeney-Lis conjecture: joint scaling limit of critical Ashkin-Teller heights and current clusters

For every fixed point of the critical Ashkin-Teller line, including the four-state Potts endpoint, every bounded Jordan domain and every admissible polygonal approximation, the height, the wired and free current-cluster collections and the distinguished boundary cluster converge jointly: the height to a GFF with the predicted coupling constant $\pi\sqrt g$, and the clusters to canonical recursive two-valued local sets of the same field, at all nesting depths, in positive-diameter Hausdorff matching topology. Not claimed: cluster area measures, magnetization fields, all spin-domain-wall curves, or metric exponents.

For every fixed point of the critical Ashkin-Teller line, including the four-state Potts endpoint, every bounded Jordan domain and every admissible polygonal approximation, the height, the wired and free current-cluster collections and the distinguished boundary cluster converge jointly: the height to a GFF with the predicted coupling constant $\pi\sqrt g$, and the clusters to canonical recursive two-valued local sets of the same field, at all nesting depths, in positive-diameter Hausdorff matching topology. Not claimed: cluster area measures, magnetization fields, all spin-domain-wall curves, or metric exponents.