Connection-probability conjecture for critical loop $O(n)$ models

Consider the loop $O(n)$ model with $n\in (0,2]$ and boundary condition $\beta\in\mathrm{LP}_N$. Let $\varsigma^{\delta}$ be the random connectivity in $\mathrm{LP}_N$ induced by the $N$ discrete interfaces in $\Omega^{\delta}$. In the critical dilute phase, $p=p_c$ and $\kappa\in (8/3,4]$ solves the relation defining $\kappa$; in the critical dense phase, $p>p_c$ and $\kappa\in [4,8)$ solves the same relation. Connection-probability conjecture. For every $\alpha\in\mathrm{LP}_N$, the conformally invariant limit is

$$\lim_{\delta\to 0}\mathbb{P}_{\beta}^{\delta}[\varsigma^{\delta}=\alpha]=\mathcal{M}_{n}(\alpha,\beta)\,\frac{\mathcal{Z}_{\alpha}^{(\kappa)}(\mathring{x}_1,\ldots,\mathring{x}_{2N})}{\mathcal{F}_{\beta}^{(\kappa)}(\mathring{x}_1,\ldots,\mathring{x}_{2N})}.$$

Here $\mathcal{M}_{n}$ is the meander matrix with parameter $n$, $\mathcal{Z}_{\alpha}^{(\kappa)}$ is the pure partition function, $\mathcal{F}_{\beta}^{(\kappa)}$ is the Coulomb gas integral, and $\mathring{x}_j=\varphi(x_j)$ under any conformal map $\varphi\colon\Omega\to\mathbb{H}$. This is the predicted connection-probability formula for multiple interfaces in critical loop $O(n)$ models; the general convergence is conjectural, although special cases such as critical Ising interfaces are known.