Metric-measure limits of subcritical FK and spanning-tree planar maps

We resolve the finite spherical cases of Gwynne and Miller's graph-metric conjecture for critical Fortuin–Kasteleyn maps at each fixed $q\in (0,4)$ and for uniform spanning-tree-decorated maps. After deterministic rescaling of graph distances, these maps, equipped with the vertex probability measure proportional to degree, converge in Gromov–Hausdorff–Prokhorov law to their ordinary unit-area Liouville quantum gravity spheres. Distances use every primal edge, and convergence holds through all positive integer edge counts.