Which 2-hyponormal 2-variable weighted shifts are subnormal?

It is well known that a 2-hyponormal unilateral weighted shift with two equal weights must be flat, and therefore subnormal. By contrast, a 2-hyponormal 2-variable weighted shift which is both horizontally flat and vertically flat need not be subnormal. In this paper we identify a large class S of flat 2-variable weighted shifts for which 2-hyponormality is equivalent to subnormality. One measure of the size of S is given by the fact that within S there are hyponormal shifts which are not subnormal.