Local cohomology of Segre product type rings

One of the main questions in the study of local cohomology of normal domains that are not Cohen–Macaulay is whether the local cohomology of the ring in degrees less than the dimension of the ring is almost zero in an integral extension of the ring. Many examples of non-Cohen–Macaulay normal domains are of Segre product type. In this paper we show that the local cohomology in degrees less that the dimension of the ring for these rings is indeed almost zero in an integral extension.