  Recent VLBI observations have resolved Sagittarius A* at horizon scales. The
Event Horizon Telescope is expected to provide increasingly good images of the
region around the Schwarzschild radius $r_S$ of Sgr A* soon. A number of
authors have recently pointed out the possibility that non-perturbative quantum
gravitational phenomena could affect the space surrounding a black hole. Here
we point out that the existence of a region around $\frac76 r_S$ where these
effects should be maximal.
