FK representation conjecture for Ashkin–Teller spins

The Ashkin–Teller model on a box $\Lambda_n$ has configurations $(\sigma_1,\sigma_2):\Lambda_n\to\{-1,+1\}^2$ with density proportional to

$$\exp\left(\sum_{x\sim y}J\left(\sigma_1(x)\sigma_1(y)+\sigma_2(x)\sigma_2(y)\right)+\sum_{x\sim y}U\sigma_1(x)\sigma_1(y)\sigma_2(x)\sigma_2(y)\right).$$

On the self-dual line, assume $J\geq0$, $J\geq U$, and

$$\sinh(2J)=\exp(-2U).$$

Let $g\in(4/3,4]$ be given by

$$g=\frac{8}{\pi}\sin^{-1}\left(\frac{\coth 2J}{2}\right).$$

Let $CLE_4^g$ denote the iterated CLE$_4$ gasket construction described in the source, let $s_C$ be the spin associated with a gasket $C$, and let $\nu(C)$ be its Minkowski content measure. FK representation conjecture for Ashkin–Teller spins. An FK representation of the continuum limit of $\sigma_1$ with wired boundary conditions is given by

$$\sum_{C\in CLE_4^g}\nu(C)s_C,$$

where the sum is taken in decreasing order of gasket diameter over all CLE$_4$ gaskets in the iteration. This conjecture proposes a continuum FK-type representation along the Ashkin–Teller critical line. The source notes that natural measures on the CLE$_4$ gasket are available or expected to agree up to constants with the Minkowski content measure, and that the conjecture was independently proposed in related work and proved for the XOR-Ising case; the general assertion here remains open.