Conjecture on the size of a critical percolation cluster in a Boltzmann triangulation

Let $T^{(\ell)}$ be a critical random Boltzmann triangulation of the $\ell$-gon, with its simple boundary colored black, and let $\mathfrak{C}^{(\ell)}$ be the boundary cluster of a critical site or bond percolation on $T^{(\ell)}$. 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. The cluster-size conjecture. The boundary cluster satisfies

$$\mathrm{v}(\mathfrak{C}^{(\ell)})\approx \ell^{7/4}.$$

This predicts that a block with perimeter of order $n^{2/3}$ can have total size comparable to the critical cluster, whose size is of order $n^{7/6}$. The exponent is also intended to determine the critical exponent for the size of the origin cluster in the UIPT.