The tenuous state of women’s basic right to make their own childbearing decisions was made clear on Monday when a federal appeals court in New Orleans heard arguments on a new abortion restriction enacted in July in Texas — one that requires doctors who perform abortions to have admitting privileges at a nearby hospital.

In October, a federal district judge, Lee Yeakel, ruled that the requirement serves no medical purpose and improperly infringes on women’s reproductive rights. A three-judge panel of the United States Court of Appeals for the Fifth Circuit overturned his injunction on the rule’s enforcement, and the Supreme Court, in an alarming 5-to-4 opinion in November, declined to upset the panel’s ruling.

The requirement immediately forced 12 of the state’s 34 abortion clinics to stop offering the procedure, according to the Texas Policy Evaluation Project, a research group, though four subsequently resumed services because doctors were able to secure hospital privileges.

Another provision of the Texas law requires abortion clinics to meet surgery-center standards — including those that perform only nonsurgical medication abortions and the safest early-stage procedures. When that provision goes into effect in September, the number of facilities performing abortions could drop steeply from the current 26 to as few as eight statewide, according to Planned Parenthood.