I don't understand the point of an investigation here. It's not like there's any mechanism whatsoever to hold an unethical justice to account, and that's by design. There is literally no ethical oversight over the Supreme Court.
> He did it to keep Roberts's vote secure
He had no idea how Roberts would eventually vote. Robert's been very much a wild-card, so this would be quite the reckless gamble. Further, generally speaking, SCOTUS doesn't really have much stake in politics, so I also very much doubt it was a Justice that leaked it. The public overestimates how much Justices actually care about the issues themselves; they mostly care about their Constitutionality (and philosophical implications therein).
I do think they actually know who leaked it, but naming and shaming would just lead to a lynch mob, political theater, and further erode trust in SCOTUS.
Sadly, I think you are missing Alito's political slant. That aspect typically makes him a super villain and, of course, guilty of all accusations.
Accusations seem to carry far too much weight in modern times...