Discrete-to-continuum quantum-disk welding conjecture for self-avoiding bubbles

For $\ell>0$, let $\overline{\mathrm{MD}_{0,2}^n}(\cdot,[\ell n^{1/2}])^\#$ and $\underline{\mathrm{MD}_{0,1}^n}([\ell n^{1/2}])^\#$ be the probability measures on the two types of quadrangulated disks with the indicated discrete boundary length, and let $\mathrm{Weld}_d^{\mathrm{bubble}}$ denote their discrete conformal welding along the matching boundary. Let $\mathrm{QD}_{0,2}(\cdot,\ell)^\#$ and $\mathrm{QD}_{0,1}(\ell)^\#$ be the corresponding fixed-boundary-length quantum-disk laws, and let $\mathrm{Weld}_c^{\mathrm{bubble}}$ denote their continuum welding. Discrete-to-continuum welding conjecture. For every $\ell>0$,

$$\mathrm{Weld}_d^{\mathrm{bubble}}\left(\overline{\mathrm{MD}_{0,2}^n}(\cdot,[\ell n^{1/2}])^\#,\underline{\mathrm{MD}_{0,1}^n}([\ell n^{1/2}])^\#\right)\xrightarrow{w}\mathrm{Weld}_c^{\mathrm{bubble}}\left(\mathrm{QD}_{0,2}(\cdot,\ell)^\#,\mathrm{QD}_{0,1}(\ell)^\#\right).$$

The convergence is in the Gromov–Hausdorff–Prokhorov–uniform topology. This asserts convergence of the discrete welding construction to continuum quantum-disk welding at fixed boundary length; it remains open in the source.