In principle, this has been tested already. The test particle is a
muon that has a well known half life as measured in the lab. It is produced in the Earth's upper atmosphere as a part of a reaction chain caused by cosmic particles.
The experimental setup is shown here:
http://hyperphysics.phy-astr.gsu.edu...iv/muonex.html
http://hyperphysics.phy-astr.gsu.edu...ativ/muon.html
Without relativistic effects, the life time of a muon is too short for it being able to reach the surface of the Earth. If you calculate the time it takes to reach the ground, it should have been decayed before it can be detected. But the fact that we do measure quite an amount of muons, it tells us that relativistic effects prolong its half life when being observed from our reference frame. If seen from the muon's perspective, the distance from its production in the upper atmosphere to the ground appears much smaller than we measure it.