Improved two-grid approximation conjecture for fractional PDE-constrained optimization

Let ${\rm dom}$ be a convex domain, let $s$ satisfy $0<s<1$, and let ${\mathcal H}_h^s$ and ${\mathcal G}_h^s$ denote the fine-grid and two-grid operators, respectively, with mesh size $h$ and regularization parameter $\beta$. Improved two-grid approximation conjecture. There is a constant $\tilde{C}_s$ independent of $h$ such that, for sufficiently small $h$,

$$d({\mathcal H}_h^s,{\mathcal G}_h^s)\leq \begin{cases} \tilde{C}_s\beta^{-1}h^{4s},&0<s<1/2,\\ \tilde{C}_s\beta^{-1}h^2,&1/2\leq s<1. \end{cases}$$

This conjecture predicts a substantially stronger two-grid, and hence multigrid, approximation estimate than the preceding theorem, with improved convergence rates suggested by the numerical experiments. Its validity is stated for convex domains and sufficiently fine meshes, but the supplied text does not indicate whether it has been proved or remains open.