On a relation between certain cohomological invariants

Let G be a group, image the supremum of the projective lengths of the injective image-modules and image the supremum of the injective lengths of the projective image-modules. The invariants image and image were studied in [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203–223] in connection with the existence of complete cohomological functors. If image is finite then image [T.V. Gedrich, K.W. Gruenberg, Complete cohomological functors on groups, Topology Appl. 25 (1987) 203–223] and image, where image is the generalized cohomological dimension of G [B.M. Ikenaga, Homological dimension and Farrell cohomology, J. Algebra 87 (1984) 422–457]. Note that image if G is of finite virtual cohomological dimension. It has been conjectured in [O. Talelli, On groups of type Φ, Arch. Math. 89 (1) (2007) 24–32] that if image is finite then G admits a finite dimensional model for image, the classifying space for proper actions. We conjecture that image for any group G and we prove the conjecture for duality groups, fundamental groups of graphs of finite groups and fundamental groups of certain finite graphs of groups of type image.