Uniform convergence of canonical discrete uniformization on equilateral approximations

We prove uniform convergence of the canonical discrete uniformization maps for every sequence of equilateral polygonal disks whose mesh tends to zero and whose marked Jordan boundaries converge uniformly. The limit is the normalized Riemann map, with convergence on the entire closed reference triangle. This proves Conjecture 7.3 of Luo, Sun and Wu for the original construction with Delaunay flips. The proof combines closed-domain compactness, marked lattice limits, a packing and resistance estimate for logarithmic cusp scales, and an envelope comparison principle, followed by a local affine energy argument.