Time for the SG to file -- I searched on the Supreme Court website for the phrase "the solicitor general is invited", there are 39 pages of results. A quick sampling indicates time to respond varies from 2 months in a few cases to 3 or 4 months in many and 6 or 7 in others.
One factor is that an invitation may be made in many cases, so the SG may have many briefs to write at once. I have to think that would play a role as well. If and when such an invite is issued in this case, I will probably try and figure out how many other requests have been made and are pending.
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