Curvature-zero propagation conjecture for eternal triod blow-ups

Let $\mathbb T^\infty_{\mathfrak t}$ be an eternal evolution by curvature of an unbounded embedded triod, curve, or curve with a single endpoint, and let its curvature be defined along each curve. Curvature-zero propagation conjecture. If the curvature is zero at some point of $\mathbb T^\infty_{\mathfrak t}$, then it is zero everywhere along the curve containing that point. These conjectures concern the structure of blow-up limits at Type II singularities. The source presents this as unresolved; proving it would provide a convexity-type property useful for excluding such singularities.