Generalized discrete holomorphicity conjecture for random cluster models

Let $q\in[0,4]$, and consider the random cluster model on lattice domains. For the Ising model, let $F(z)$ denote the function in Proposition~, which is discrete holomorphic and satisfies the discrete analogue of the Riemann boundary value problem, and let $F_j$ be its counterpart on lattice domains converging in the Carathéodory sense to a continuum domain. Generalized discrete holomorphicity conjecture. Proposition~ (with an appropriate, possibly approximate, discrete analyticity) and Theorem~ hold for all values of $q\in[0,4]$. The result is proved in the Ising case $q=2$; extending the discrete analyticity and scaling-limit argument to the remaining values of $q$ was left open.