Conjectured Brownian-sphere limit for planar sparse random maps

Let $\mathrm{M}_n(\mathrm{f}_n,0)$ be a random plane map with $\mathrm{f}_n$ faces and $n$ edges. Assume

$$\mathrm{f}_n\to\infty,\qquad n^{-1}\mathrm{f}_n\to0.$$

Brownian-sphere conjecture. The rescaled maps

$$\sqrt{6}\,\frac{3^{1/4}}{1+2\sqrt{3}}\mathopen{}\mathclose\bgroup\originalleft(\frac{\mathrm{f}_n}{n^2}\aftergroup\egroup\originalright)^{1/4}\mathrm{M}_n(\mathrm{f}_n,0)$$

converge in distribution to the Brownian sphere in the Gromov--Hausdorff topology. The preceding discussion motivates this scale from the core and tree diameters, but the paper explicitly refrains from making the claim precise and presents it as a belief.