Conjecture on the size of the largest critical percolation cluster in a uniform triangulation

Let $T_n$ be a uniform triangulation with $n$ faces, and perform a critical site or bond percolation on $T_n$. For random variables, write $X_n\approx n^{\alpha}$ when, for every $\epsilon>0$, the probability that $n^{\alpha-\epsilon}<X_n<n^{\alpha+\epsilon}$ tends to $1$ as $n$ tends to infinity. Largest-cluster conjecture. If $\mathfrak{C}_{\mathrm{max}}$ denotes the largest black cluster in the percolated triangulation, then

$$\mathrm{v}(\mathfrak{C}_{\mathrm{max}})\approx n^{7/8}.$$

The prediction follows the heuristic that a critical cluster of size $n$ is contained in a triangulation of size of order $n^{8/7}$; it further asserts that the initial cluster has a positive chance of being the largest cluster after the filling operation. The claim remains conjectural.