The correct answer is Felix Frankfurter wanted to outlaw segregation, so he asked that the case be reargued so that he had time to convince more justices and build consensus
He believed that there was a need to solve it with "all deliberate speed" which means that desegregation had to be rapidly implemented and not dragged on. This plan of his backfired and the thing did drag on with many schools in the South waiting for as much as 15 years after the court decision before finally desegregating.