We introduce a new family of classical r-matrices for the Lie algebra sln that lies in the Zariski boundary of the BelavinâDrinfeld space M of quasi-triangular solutions to the classical YangâBaxter equation. In this setting M is a finite disjoint union of components; exactly Ï(n) of these components are SLn-orbits of single points. These points are the generalized CremmerâGervais r-matrices ri,n which are naturally indexed by pairs of positive coprime integers, i and n, with i<n. A conjecture of Gerstenhaber and Giaquinto states that the boundaries of the CremmerâGervais components contain r-matrices having maximal parabolic subalgebras pi,nâsln as carriers. We prove this conjecture in the cases when nâĄÂ±1 (mod i). The subprime linear functionals fâpi,nâ and the corresponding principal elements Hâpi,n play important roles in our proof. Since the subprime functionals are Frobenius precisely in the cases when nâĄÂ±1 (mod i), this partly explains our need to require these conditions on i and n. We conclude with a proof of the GG boundary conjecture in an unrelated case, namely when (i,n)=(5,12), where the subprime functional is no longer a Frobenius functional.