In his concurring opinion, Justice Kavanaugh said it was likely unconstitutional (but apparently not obviously enough to enjoin it) [1]. So it's going into effect, then the lawsuit follows.
Similar laws in California, Arkansas and Ohio were all found unconstitutional, so I am hopeful. That said, these were all district court decisions, and all of them are being appealed. When they lose on appeal, they go to the Supreme Court for (hopefully) the final smack-down.
Interestingly, reading the summary MS HB1126 [2], this law is doing two things. It regulates companies and defines crimes.
States are allowed to set their own criminal codes. If Mississippi drops the mandate part and passes a new law that simply defines certain things as crimes with corresponding penalties, that law would probably be constitutional.
[1] https://www.supremecourt.gov/opinions/24pdf/25a97_5h25.pdf