String links with the same closure and group diagrams

We give a direct proof that if two string links have isotopic closures, then there is a braid-special isomorphism between their n-level group diagrams, for every n >= 2. In the case of link-homotopy, we give an alternative proof to our previous result that there is a braid-special isomorphism between the group diagrams for the homotopy classes of two string links if and only if they have link-homotopic closures.