As observers, the detectors would see precisely what we expect if we calculate the distance using the detector's proper time when compared with the distance we experimentally measure.
The half life of a static tetraneutron is (theoreticaly) 5E-22s.
If you are at a laboratory and the tetraneutron is moving at relativistic speed, the apparent half life in the laboratory frame will increase substantially.
To calculate the distance that the tetraneutron will travel you must multiply the velocity x the apparent half life. I.E
d ~= v * 1/(1-v^2/c^2) * t_hl
where t_hl = 5E-22s = the half life of the tetraneuton
not the classic version that is
d_clasic_wrong = v * t_hl
For a more detailed explanation, with numbers and graphic you all can see the analogue experiment with muons: http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/muon.html
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Anyway, IIRC a half life of 5E-22s is still very small. IIRC you can use a bubble chamber or something similar to see the trajectory of the particles. You can only see a collision and examine the debris and you will note that when the collision has some specific energy you will get an unexpected excess of collisions. You only can examine the direction and energy of the debris to make a good guess of the intermediate steps of the collision and discover that for a very short time you had a tetraneutron.
In retrospect perhaps the parent comment was saying theoretically it could travel any distance which is true, but doesn't really help in estimating whether it could clear nuclear radii or not.