One can define an analogous version of the curve shortening flow for polygons in $\mathbb R^2$, namely defined by the differential equation $\dot{p_i}(t)=\frac{v_i(t)}{|v_i(t)|^2}$, where $p_i$ is the $i$-th vertex of the polygon and and $v_i$ is the vector that goes from $p_i$ to the circumcenter of the triangle defined by $p_{i-1}$, $p_i$ and $p_{i+1}$.

The Gage-Hamilton-Grayson theorem states that simple curves remain simple under the curve-shortening flow.

Does this still hold for polygons under this analogous flow?