Sharp entropy constant conjecture for ancient curve shortening flows

Let $M_t^1 oindent\subset \mathbb{R}^N$ be an ancient curve shortening flow, and let $\lambda(M_t)$ denote its entropy. Suppose that $M_t$ does not lie in a lower-dimensional Euclidean subspace.

Sharp constant conjecture. The sharp value of the universal constant $C_1$ in the Colding–Minicozzi codimension bound is

$$C_1=\frac{2}{\lambda(S^1)}.$$

In particular,

$$\sup_t\lambda(M_t)\geq\frac{N}{2}\lambda(S^1).$$

This conjecture proposes sharpness of the entropy-dependent Euclidean-subspace dimension bound for ancient one-dimensional mean curvature flows; the torus curve examples and the shrinking circle give the matching lower bound currently described in the source.