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My application is basically complete except for 1 LOR. (I have 2 already). The Career services are telling me to apply now and that when the last LOR comes in, they will either upload it to Oscar or separate mail to the judge for paper apps.

Is that ok? I think the urgency is that some judges are reviewing apps even before the deadline. But I am afraid that since its an "incomplete app", they would hate it.
Any current clerks that can speak to this?

I'm not a current clerk but I would just send it as is unless the judge has specifically noted (in OSCAR) that they only want complete applications. No judge is going to hate you because the recommendations came in seperately.

Yes, this is fine. It's extremely common.

Love, current clerk.