Optimality conjecture for geometric polynomial approximation of circular arcs

Let $\bfm{c}$ be a circular arc and let $\bfm{p}_n$ be a $G^k$ geometric approximant of degree $n$. Choose a polynomial $p_{2n,k}^*$ minimizing

$$\|p_{2n,k}\|=\max_{t\in[-1,1]}\left|(1-t^2)^{k+1}q_{n,k}(t)\right|,$$

where $q_{n,k}$ has degree $2n-2k-2$, and let $\bfm{b}_j$, $j=0,1,\dots,n$, be the control points determined by this choice. Optimality conjecture. The best $G^k$ geometric approximant $\bfm{p}_n$ of the circular arc $\bfm{c}$ according to the error measure $\psi_{n,k}$ arises from the choice $p_{2n,k}^*$ determining $\bfm{b}_j$, $j=0,1,\dots,n$, with minimal $\left|C(\bfm{b}_0,\dots,\bfm{b}_n)\right|$. The conjecture would establish the proposed framework as an optimal construction for circular-arc approximation and would extend the paper's known optimal results for earlier cases while covering the proposed $G^0$ cubic and $G^1$ quartic approximants. The supplied text gives no resolution, so its status remains open.