[math-fun] Special pythagorean triples

Someone on quora asked if there are Pythagorean triples of the form (x^2-1)^2 + (y^2-1)^2 = (z^2-1)^2, other than {x,y,z} = {10,13,14} and {265,287,329}. From a probabilistic standpoint it seems unlikely (and verified for z < 2^15), but is there a proof? Of course, without the -1s, there are no solutions--it's just n=4 of FLT.