Analytic graph conjecture for the Kuramoto global attractor

Let $q_h$ be the heteroclinic solution of the Kuramoto equation connecting the incoherent equilibrium to the synchronized equilibrium, and let $W^{\frac{1}{2\pi}}$ denote the global attractor, represented as a graph $\Phi_\mathcal H$ over the disk in the first Fourier coefficients. Analytic graph conjecture. The global attractor is an analytical graph on the disk $\{(x_1,y_1):x_1^2+y_1^2<r\}$. The preceding numerical observations suggest analyticity of the graph function and its Fourier-coordinate functions, but the claim is presented as part of a partially proved, partially numerical proof sketch.