Convergence conjecture for the normalized CR Yamabe flow

Let $(M^3,J,\mathring{\theta})$ be a closed spherical CR $3$-manifold with positive Tanaka-Webster scalar curvature and vanishing torsion. Consider the normalized CR Yamabe flow starting from $\mathring{\theta}$. Convergence conjecture. As $t\to\infty$, the solution converges smoothly to a unique limit contact form of positive constant Tanaka-Webster scalar curvature and vanishing pseudohermitian torsion. This would establish asymptotic convergence of the normalized flow in this spherical, torsion-free setting; the surrounding discussion indicates that such convergence is otherwise widely open even on closed pseudohermitian $3$-manifolds.