Of course, if you apply to the Copyright Office for special handling and use the words “ongoing infringement,” they will bring your application to the top of the pile and get the registration, if granted, to you in a few days or weeks rather than a few months.
However, it would be delightful if the Nine Wise Ones decide that filing for copyright registration is sufficient to proceed to litigation under 17 USC 411(a); that would ease the burden on the litigants to either wait for the Copyright Office to operate in due course or pay the extra fees to get them to operate more quickly than due course. It would ease the burden on the Copyright Office to handle “rush” jobs that are waiting at the courthouse door. Of course, the defense bar would disagree with me….
I’ve been known to take my own sweet time with the pre-filing settlement negotiations while I await the Copyright Office’s action in registering a client’s infringed work. That sometimes works better than litigation for all parties, so I’ve never actually found that the Copyright Office has held up anything I desperately wanted to do, especially since the fact of removing the infringing item does not negate the fact of the infringement for which my plaintiff is entitled to remedy.