Grid homotopic curve shortening approximates affine curve-shortening flow

Let $\delta$ be an initial curve, and let $\gamma_0$ be the shortest curve homotopic to it among curves avoiding the obstacle set $P_n=(\mathbb Z/n)^2$. For $t>0$, write $\delta'=\delta(t)$ for the curve obtained from $\delta$ under affine curve-shortening flow (ACSF), whenever it is defined. Let $c_{\mathrm g}\approx 1.6$ be a constant, set

$$m=\lfloor c_{\mathrm g}tn^{4/3}\rfloor,$$

and let $\gamma_m=\operatorname{\mathrm HCS}_P^{(m)}(\gamma_0)$ be the result of $m$ homotopic curve-shortening iterations. Grid approximation conjecture. There exists a constant $c_{\mathrm g}\approx 1.6$ such that, for every fixed $t>0$ for which $\delta'$ is defined, the Fr\e9chet distance between $\gamma_m$ and $\delta'$ tends to $0$ as $n\to\infty$. This conjecture formalizes the experimentally observed connection between HCS on increasingly fine uniform grids and ACSF, generalizing the cited grid-peeling conjecture; its resolution is not given in the source.