Who assigned this and why? (Update)

Who assigned Dobbs to Alito--the Chief or Thomas?

Assume the following at conference: Thomas, Alito, Gorsuch, Kavanaugh, and Barrett want to overrule Roe, declare the MS law valid, and enter judgment for the state; the Chief wants to declare the MS law valid as not imposing an undue burden  and enter judgment for the state. Who is the senior-most Justice in the majority? What is the "majority" when in Conference and before any opinions have been written--is it a majority for the judgment ("the law is constitutionally valid, plaintiffs lose, state wins") or is it majority for a rationale or an answer to a QP (Roe/Casey are overruled)? If the former, the Chief keeps the assignment; if the latter, Thomas gets the assignment.

I raised this question (without a satisfactory answer) over the now-meaningless June Medical, where four Justices (Breyer for Ginsburg, Sotomayor, and Kagan) declared the law invalid by balancing burdens and benefits under Casey and the Chief agreed the law was invalid but on the logic of WWH and considering only the burdens; did the Chief assign or did Ginsburg? Does the assignment work differently when there is a majority for a result but not for a rationale, as in June, as opposed to where there is a majority for a rationale plus extra votes for the result?

My best guess is Thomas assigned it. If so, I am impressed (and a bit curious) that he chose not to keep it for himself. Casey was decided during Thomas's first Term, meaning he has been waiting his entire time on the Court for this opportunity.

If Roberts assigned it, the choice of Alito creates all sorts of Kremlinology. If Roberts  (presumably) wanted to make the least noise, he would not have assigned it to Alito, knowing the likely tenor of the opinion. Or he assigned it to Alito intentionally, knowing he might draft an opinion (what my colleague called a "nuclear bomb overruling") that might scare off Kavanaugh or Barrett. In which case the "conservative leaker" theory makes sense as a counterpunch to that. Anyway, I doubt anyone thinks this way, which is why I believe the relevant majority was to overrule Roe and Thomas gave the opinion to Alito.

Update: This question was raised on the Con Law Prof listserv. No one knew for sure, although one former clerk says his understanding is that the majority is for the bottom-line disposition. This make some sense, the person argued, because some justices only have identified a conclusion but not a reason at conference. Alternatively, many cases may have a bottom-line majority but competing reasons, none garnering the initial support of any 5; the only way to identify a seniormost-in-the-majority is to go by majority for the judgment, meaning the Chief assigned Dobbs. It also would follow that the Chief, rather than Ginsburg, gave Breyer June Medical.