Uniqueness of near-critical Ising scaling limits

Let $\lambda\neq 0$ be fixed and let $(\omega_n,\sigma_n)$ be an FK-Ising coupling of an Ising model at $\beta=\beta_c+\frac{\lambda}{n}$ in $[-1,1]^2\cap\frac{1}{n}\mathbb{Z}^2$, with any of the boundary conditions $a)$, $b)$, or $c)$ listed at the beginning of the section. The pair may be viewed as a random point in several topological spaces: FK clusters with prescribed colours under the Schramm–Smirnov quad-topology or a Hausdorff-like topology on coloured loops; the spin configuration as a random magnetic field $\Phi_n$ in a negative-index Sobolev space; or as a collection of loops separating $+$ and $-$ clusters. Uniqueness conjecture. Under each of these topologies, $(\omega_n,\sigma_n)$ should converge to a unique limiting $\lambda$-near-critical scaling limit. This concerns the existence and uniqueness of near-critical continuum limits, which is not known in general, although several corresponding critical scaling limits and topological equivalences are established.