Constant-curvature convergence conjecture for normalized Ricci flow on trees

Let $T=(V,E)$ be a tree with an initial metric $w\in\mathbb{R}_{>0}^{E}$, and let the metric $w(t)$ evolve under the normalized Ricci flow. The flow converges to a metric with constant curvature if there exists a metric $w(\infty)\in\mathbb{R}_{\geq 0}^{E}$ such that

$$\lim_{t\to\infty}w(t)=w(\infty)$$

and, writing $E^{+}=\{e\in E(T):w_e(\infty)>0\}$, the curvature under $w(\infty)$ is the same on every edge in $E^{+}$. Constant-curvature convergence conjecture. For every tree with an initial metric, the normalized Ricci flow converges to a metric with constant curvature. This conjecture proposes a global asymptotic property of normalized Ricci flow on trees; the supplied source does not establish whether it is known or remains open.