Scaling-limit conjecture for the interface length in critical Ising triangulations

Let $(\mathfrak{t},\sigma)$ be sampled under $\mathbb{P}_{p,q}$, and let $\eta_{p,q}$ be the length of its leftmost interface. Let $E$ be the expectation under $\mathbb{P}_\infty$ of the random variable describing the contribution of one peeling step to the total interface length, taking values in $\{0,1,\eta_{k,1},1+\eta_{k,1}:k=0,1,\dots\}$. Scaling-limit conjecture. As $p,q\to\infty$ with $q/p\to\lambda\in(0,\infty)$, one should have

$$\mathbb{P}_{p,q}(\eta_{p,q}>tp/\mu)\longrightarrow \int_{t/E}^{\infty}(1+s)^{-7/3}(\lambda+s)^{-7/3}\,ds.$$

The conjecture identifies the leftmost interface length with the hitting-time scaling limit established for the peeling exploration, up to the mean contribution $E$ of a peeling step. Its status is not resolved in the supplied source.