Ah, now I see your problem : you think that there is such a thing as reciprocal length contraction and you demand an experiment that proves it.
OK, this is one of the weaker of the two "paradoxes", is the poor cousin of Dingle's paradox. Anyway, reciprocal length contraction does not exist. It is provable in about 1 page of formulas.
Suffice to say that SR DOES NOT CLAIM that reciprocal length contraction exists, quite the contrary, it refutes such a notion. Neither does "mutual time dilation" exist (2 pages of calculations). Happy now?