Conformal-radius convergence-rate conjecture for the Method of Fundamental Solutions

Let $Omega$ be a simply connected domain with analytic boundary, and let $\rho_z$ denote the conformal radius of $z$ with respect to $Omega$. Place the MFS points equally spaced in conformal angle at conformal distance $R>1$ around $Omega$. Let $t$ be the MFS error, and let $\rho>1$ be the conformal radius of the closest singularity of the analytic continuation of $u$. Conformal-radius convergence-rate conjecture. There is a constant $C$, depending possibly on $Omega$, $k$, $R$ and $v$ but not on $N$, such that

$$t \le \begin{cases} C\rho^{-N/2},& \rho<R^2,\\ CR^{-N},& \rho>R^2.\end{cases}$$

Furthermore, if $u$ continues to an entire function, the latter estimate holds for every $R>1$. This conjecture extends the corresponding convergence-rate result from the unit disk to general analytic domains; the case $\rho=R^2$ is deliberately excluded because numerical evidence does not determine whether an additional algebraic factor is required.