Conformal-radius coefficient-growth conjecture for the Method of Fundamental Solutions

Let $Omega$ be a simply connected domain with analytic boundary, and let $Gamma_R=\{z:\rho_z=R\}$ be the curve of points with conformal radius $R>1$. Choose the MFS charge points equally spaced in conformal angle on $Gamma_R$. Let $\rho$ be the conformal distance of the closest singularity of the analytic continuation of $u$, let $t$ be the MFS error, and let $\boldsymbol{\alpha}$ be the vector of coefficients of the MFS basis functions that minimizes $t$. Conformal-radius coefficient-growth conjecture. There are constants $C$ independent of $N$ and $\gamma>1$ such that

$$|\boldsymbol{\alpha}|\geq C\gamma^N$$

if and only if $R>\rho$. This conjecture generalizes the observed coefficient-growth dichotomy from the disk to general analytic domains: exponential growth is predicted precisely when the charge-point curve lies beyond the nearest singularity.