From the article: "Incestuous marriages are forbidden in all states, in part due to fears about genetic mutation."
In Lawrence, the rights of two individuals to enter into a private and intimate relationship were defined. Incest laws protect the rights of innocent individuals, the possible offspring of the relationship.
In some States, in order for first cousins to marry, they must first prove that there are no possibilities of an issue from the marriage.
The SCOTUS decision in Lawrence vs. Texas addressed a very specific point, it did not create the sort of slippery slope we are so fond of predicting here in FR.
Yes they did say, and said explicitly, it does not create that sort of slippery slope but the reasoning, those firm principles upon which it rests are not terribly firm. Ultimitately it creates a constitution that says whatever the judge says that it says.
Heck you probably could have overturned the Texas law based on 14th Ammendment but they over reached and that will come back to bite us.
So they will go with "compelling state interest" as their reasoninh, which means whatever 5 of 9 say it does.