They lose. The issue is too political to be treated abstractly and objectively. Their argument will be that the tenth amendment is clear. All powers not contitutionally delegated to the fedral government are reserved by the states. The constitution doesn't mention secession, and certainly doesn't delegate the issue to the federal government. Nor does the constitution, unlike the earlier Articles of Confederation, declare the union to be perpetual. Therefore, secession is reserved to the states. End of story. And you know what? Legally speaking, I'm 100% convinced they would be correct.
Doesn't matter. No Supreme Court Justice will want to be known as the man (or one of the men) who "broke the Union". Some legal fiction will be brought up to deny that secession may take place. Perhaps they'll argue that since the constitution replaces the Articles (which were explicitly defined as perpetual), that perpetuity is implicitly carried on into the constitution. Something like that. Legally speaking, it would be a travesty. But then, so is OTL. In every country, in every era. (And morally speaking, of course, anyone can see that a legal travesty that results in an end to chattel slavery may be excused.)
Bottom line: the South loses. Ironically, if that defeat breaks the support for secession (which it may well do), that very much delays the abolition of slavery. So we still get a legal travesty... but no undisputed Moral Good that can justify it. (On the other hand: no Civil War. Which reduces us to asking: is preventing the many deaths of that conflict worth prolonging chattel slavery by - in all likelihood - decades? Interesting subject matter for those of us in the ethics department, but not a debate to be held here, I think.)