No, they don’t bring cases up to conference based on courtesy.
***I wish we could get the constitutional scholars and lawyers on this site to weigh in on that statement. It strikes me as perfectly sensible, but all I hear is hemming & hawwing when I ask about it.
I’m not a lawyer but work with/for one who’s gone to the SCOTUS and obtained certiorari. Don’t know the inner workings of the Court, but they receive 6-8K petitions a year and accept fewer than 100 for argument; some are granted certiorari but sent back to the court below without a SCOTUS hearing. The odds for any petitioner are poor.
This is my layman’s view of what happens: I think all petitions are initially read by the justices’ clerks; most are rejected. Then the lawyer/petitioner can go back to a single Justice for review, based on the Justice’s area of jurisdiction. Each Justice is assigned certain appellate areas (circuits). If that is rejected, the lawyer/petitioner can choose one more Justice whose history might show a sympathy to the issues raised for review. Then, if that Justies DOES find the issues to be worthy of SCOTUS consideration, s/he brings to Conference. At Conference they discuss all the cases and decide which to grant certiorari.