On the strong equality between supercompactness and strong compactness

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if $V \models$ ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension $V[{G}] \models$ ZFC + GCH in which, (a) (preservation) for $\kappa \le \lambda$ regular, if $V \models {}$ “$\kappa$ is $\lambda$ supercompact”, then $V[G] \models {}$ “$\kappa$ is $\lambda$ supercompact” and so that, (b) (equivalence) for $\kappa \le \lambda$ regular, $V[{G}] \models {}$ “$\kappa$ is $\lambda$ strongly compact” iff $V[{G}] \models {}$ “$\kappa$ is $\lambda$ supercompact”, except possibly if $\kappa$ is a measurable limit of cardinals which are $\lambda$ supercompact.