I don't think they're going to put everything on hold until that seat is filled; if nothing else, the lack of movement toward confirming or denying a nominee means that they would become grossly backlogged by doing so.
There are very few cases where Supreme Court action is mandatory; the Supreme Court choosing not to take cases won't result in them being backlogged, it will result in lower court decisions being let stand (forming only local, rather than national, authoritative precedent) because the Supreme Court choose not to hear appeals.
Additionally, a 4-4 vote has exactly the same result as them not taking up the case at all, so it is really just a waste of time for them to take up cases that are expected to result in a 4-4 vote.
