Uniqueness conjecture for curve shortening flow and blooming at infinity

Let $(\mathbb{R}^2,g)$ be the plane equipped with a complete smooth $O(2)$-invariant metric with non-positive curvature. Curve shortening flow is unique on $(\mathbb{R}^2,g)$ when any two uniformly proper solutions with the same initial data have the same image wherever both solutions are defined. The metric $g$ allows blooming at infinity if there exist $T\in(0,\infty)$ and a solution $R:(0,T)\to(0,\infty)$ of

$$\frac{\partial R}{\partial t}(t)=-\frac{\partial\phi}{\partial r}(R(t))$$

such that $R(t)\to\infty$ as $t\searrow0$, where $g=dr^2+e^{2\phi(r)}d\theta^2$ in polar coordinates; otherwise, $g$ does not allow blooming at infinity.

Uniqueness conjecture. CSF is unique on $(\mathbb{R}^2,g)$ if and only if $g$ does not allow blooming at infinity.

The conjecture asserts that, within this class of complete rotationally symmetric non-positively curved metrics, blooming at infinity is precisely the obstruction to uniqueness for CSF with arbitrary initial data. The source states this tentatively and does not give a resolution; uniqueness is proved in the special case of solutions starting from a radial geodesic when blooming at infinity is absent.