Fiber cones with almost maximal depth

Let A be a noetherian local ring with dimension d and I be an ideal of A. Let E = (En)(n >= 0) be a good 1-filtration of submodules of an A-module E. Let H be an ideal of A containing I and F-H(E) = circle plus(n >= 0) E-n/HEn. Assume that E is a Cohen-Macaulay module with lambda(E/IE) finite and Ann(E) = 0, and let J be a minimal reduction of I. In this paper we give conditions on lambda(E-n boolean AND JE/JE(n-1)) and lambda(HEn boolean AND JE/JHE(n-1)) so, that F-H(E), has depth of at least d - 1.