In Gabai's paper (Knot Theory and Manifolds Lecture Notes in Mathematics Volume 1144, 1985, pp 14-17 An internal hierarchy for 3-manifolds) he considers sutured manifold decompositions of Haken $3$--manifolds. Now, one consequence of this is the theorem stated in the paper, where he provides a brief outline showing that if $M$ is a Haken $3$--manifold, with hyperbolic structure of finite volume ($M$ is compact irreducible with torus boundary components) then there exists a decomposition of $M$ into manifolds $(M_{1},\ldots,M_{k})$ with $M_{k} = T^{2}\times I$ such that for each $i$, $M_{i}$ is hyperbolic.

Now suppose I have a hierarchy $\mathcal{H}$ for a Haken $3$--manifold $M$, where $M$ is compact,irreducible and has torus boundary. Let $\mathcal{H} = (M_{1},\ldots,M_{k})$ and suppose manifold $M_{k} = T^2\times I$. Would any one know the conditions required of $\mathcal{H}$ to satisfy Gabai's theorem mentioned above. I guess another method of looking at this is to ask for the obstruction to proving the converse of Gaba's theorem?

PS. As Neil has suggested I also assume that $M$ is atoroidal (required in the context of Thurston and hyperbolization).