Uniqueness conjecture for curve shortening flow on the flat plane

Let $\mathbb{R}^2$ be the flat plane, and let CSF denote curve shortening flow. We say that CSF is unique when any two uniformly proper solutions with the same initial data have the same image wherever both solutions are defined.

Uniqueness conjecture. CSF is unique on the flat plane.

This is presented as a well-known conjecture motivated by the analogy between curve shortening flow and the heat equation, for which non-uniqueness can occur on noncompact domains. The source does not state a resolution.