I may be wrong, but if one finds "pseudo smooth" relation, one may add (the ideal) "u" to the (algebraic) factor base. Since it is with even exponent, it seems that no additional algebraic relation is needed to construct an algebraic square. Since I don't quite understand all the stuff, not all "u"s may be good, but there is chance some "u" are good.
btw, this approach works for the quadratic sieve, but Pollard/Schnorr's method is not applicable in this case.
