Why doesn't NASA launch a probe into Earth's orbital wake?

We dunno if we've cleared our own neighborhood. We just assume so.

As seen from the north solar plane the Earth orbits counterclockwise around the sun and for decades there have been comic book legends and stories about a parallel Earth that 'reputable' scientists disclaim by virtue of their degrees based mostly on theory.

To settle the argument, and it would be scientifically interesting to see if there is at least an asteroid or two sharing the orbit with Earth, why not launch a probe to fly back over the orbital path the Earth has just traveled, and we meet and recover the probe when Earth and the probe have traveled halfway around the sun going in opposite directions.

We may find some evidence out there to give the IAU cause to rethink the planet Pluto's status.