To be precise "The petition for a writ of certiorari before judgment is denied." The case is still in progress in the Tenth Circuit, (it's 23-4042 on their docket) and Brunson took the quite extraordinary step of petitioning the Supreme Court to step in before the Tenth Circuit has ruled. That's what the "before judgment" part is about. Scotus is generally very reluctant to grant cert before judgment - it is an appellate court and there's nothing to appeal until the lower court has ruled - so it is not at all surprising that this petition was denied.

So what happens next? The Tenth Circuit will eventually rule and this ruling will almost certainly go against Brunson. At that point Brunson will be able to appeal to the supreme Court arguing that Tenth Circuit got it wrong. Note that this appeal will not be arguing the about facts of the case, but whether the Tenth properly applied the law to the facts as presented in the lower courts.

Assuming that Brunson does appeal (as I expect) the appeal will appear on the Court docket with a new docket number, which means only that the right forms were mailed to the right address. There is no process for rejecting the appeal at that point. Eventually the appeal will get to the top of the stack and will be scheduled for "conference", meaning that court will decide whether they want to hear the case or not. That decision will be communicated in the next set of court orders which will either be a denial of cert so the Tenth Circuit decision stands, or scheduling the case to be heard sometime this fall when the 2023 session has started.

Me, I would bet very long odds that the Court chooses to let the eventual Tenth Circuit decision stand.