To get more statistical significance for the suspected flyby-anomaly (or to refute it), it would be desirable to track as many hyperbolic earth flybys to within 1mm/s as we can get. Can we track natural objects approaching/leaving earth with this accuracy by long-term observations?
Laser tracking of manmade objects may be easier. Think of a swarm of passive reflectors, spring-emitted way before perigee of an earth gravity-assist, so they pass with different distance and leave in different directions. But I fear that we will never see such a maneuver with a significant plane change, because missions rarely ever leave the ecliptic. With the observations so far, in- and outbound declination seems significant, especially going from high declination back to low would possibly give insight. Will the effect be reversed?
I doubt we will ever see a mission dedicated to this effect, so studying it with natural objects would be useful.