Ashkin–Teller magnetisation-field decomposition conjecture

Let the Ashkin–Teller magnetisation field be defined on the critical line, and consider the continuum iterations corresponding to the $+$ and free boundary conditions. Let the relevant two-valued sets be obtained from the current decomposition by replacing every occurrence of $\sqrt 2$ with $\sqrt g$, where $g$ is the critical-line parameter.

Ashkin–Teller magnetisation-field decomposition conjecture. The statement of the main magnetisation-field decomposition theorem holds for the Ashkin–Teller magnetisation field at every point of the critical line under the analogous $+$ and free iterations, with every explicit occurrence of $\sqrt 2$ replaced by $\sqrt g$.

The conjecture predicts a geometric decomposition of the continuum Ashkin–Teller magnetisation fields. The relevant fields are proved to be well defined in the continuum in the non-four-state-Potts case, but the asserted convergence and decomposition remain conditional on the preceding current and height-field conjectures.