Deficiency, relation gap and two-dimensional groups

Let G be a finitely presented, residually finite group and let d(G) denote the deficiency of G. Assume that every subgroup H of finite index in G satisfies d(H) - 1 = |G:H| (d(G) - 1). We conjecture that G has a two-dimensional finite classifying space K(G, 1). This conjecture is motivated by an open question about the deficiency gradient of groups and their L2-Betti numbers. In this note, we relate this conjecture to the relation gap problem for group presentations. We verify the pro-p version of the conjecture, as well as its higher dimensional abstract analogs.