On the mutual exclusion of variationality and size consistency

Why do variational electron-correlation methods such as truncated configuration-interaction methods tend to be non-size-consistent (non-size-extensive)? Why are size-consistent (size-extensive) methods such as Moller-Plesset perturbation and coupled-cluster methods non-variational? We conjecture that the variational and size-consistent properties are mutually exclusive in an ab initio electron-correlation method (which thus excludes the Hartree-Fock and density-functional methods). We analyze some key examples that support the truth of this conjecture.