Explicit complete Lyapunov function conjecture for the synchronisation map on the torus

Let $G$ be the synchronisation diffeomorphism of the torus considered in the paper, and let $S$ and $R$ be the open basins of attraction of its corresponding asymptotically stable fixed points. Define the continuous function $\mathcal{L}:\mathbb{T}^{2}\to\mathbb{R}$ by

$$\mathcal{L}(x,y)=\begin{cases} \left(x-\frac{2\pi}{3}\right)^2+\left(y-\frac{4\pi}{3}\right)^2-\left(x-\frac{2\pi}{3}\right)\left(y-\frac{4\pi}{3}\right),&\text{if }y\geq x,\\ \left(y-\frac{2\pi}{3}\right)^2+\left(x-\frac{4\pi}{3}\right)^2-\left(y-\frac{2\pi}{3}\right)\left(x-\frac{4\pi}{3}\right),&\text{if }y<x, \end{cases}$$

where $(x,y)\in[0,2\pi[\times[0,2\pi[$. Complete Lyapunov function conjecture. The function $\mathcal{L}$ is a complete Lyapunov function on $\mathbb{T}^{2}$ for $G$ in the sense of Conley, possessing the required properties on $S$ and $R$.