The case is Holt v. Hobbs; here’s the decision below (some paragraph breaks added):

In this action challenging the Arkansas Department of Correction (ADC) grooming policy under the Religious Land Use and Institutionalized Persons Act (RLUIPA), 42 U.S.C. § 2000cc-1(a)(1)-(2), inmate Gregory Holt (also known as Abdul Maalik Muhammad) appeals the district court’s order dismissing his action after an evidentiary hearing.

In his complaint and motion for a preliminary injunction and temporary restraining order, Mr. Holt asserted that one of his fundamentalist Muslim beliefs was that he must grow a beard, but defendants substantially burdened his ability to practice his religion by enforcing ADC’s grooming policy, which allowed trimmed mustaches but otherwise no facial hair, with quarter-inch beards permitted only for a diagnosed dermatological problem. Mr. Holt sought permission to maintain a half-inch beard as a compromise position, to balance his religious beliefs with ADC’s security needs.

The district court initially granted temporary injunctive relief. The court vacated its order and dismissed the complaint, however, after the hearing produced evidence that Mr. Holt had a prayer rug and a list of distributors of Islamic material, he was allowed to correspond with a religious advisor, and he was allowed to maintain the required diet and observe religious holidays; that the grooming policy helped prevent inmates from concealing contraband, drugs, or weapons; that an inmate who grew a beard could change his appearance quickly by shaving; that affording special privileges to an individual inmate could result in his being targeted by other inmates; and that prison officials believed the grooming policy was necessary to further ADC’s interest in prison security….

[W]e conclude that defendants met their burden under RLUIPA of establishing that ADC’s grooming policy was the least restrictive means of furthering a compelling penological interest, see Fegans v. Norris, 537 F.3d 897, 903 (8th Cir. 2008) (absent substantial evidence in record indicating that response of prison officials to security concerns is exaggerated, courts should ordinarily defer to their expert judgment in such matters), notwithstanding Mr. Holt’s citation to cases indicating that prisons in other jurisdictions have been able to meet their security needs while allowing inmates to maintain facial hair, see id. at 905 (although prison policies from other jurisdictions provide some evidence as to feasibility of implementing less restrictive means of achieving prison safety and security, it does not outweigh deference owed to expert judgment of prison officials who are more familiar with their own institutions).

The case was brought under the “institutionalized persons” portion of the Religious Land Use and Institutionalized Persons Act, 42 U.S.C. § 2000cc-1 (enacted in 2000):

No government shall impose a substantial burden on the religious exercise of a person residing in or confined to an institution [defined to include prisons and jails, as well as nonpenal institutions such as institutions for the mentally ill], even if the burden results from a rule of general applicability, unless the government demonstrates that imposition of the burden on that person —

(1) is in furtherance of a compelling governmental interest; and

(2) is the least restrictive means of furthering that compelling governmental interest.

This section applies in any case in which — … the substantial burden is imposed in a program or activity that receives Federal financial assistance ….

[§ 2000cc-3(d):] This chapter shall be construed in favor of a broad protection of religious exercise, to the maximum extent permitted by the terms of this chapter and the Constitution.

There seems to be disagreement among federal circuit courts on how to apply the statute, and the Supreme Court generally sees such a “circuit split” as a strong reason to agree to review the case. The decision to hear the case is thus not a very strong signal about whether the Court disagrees with the court below.

Note that the Court has unanimously rejected the argument that RLUIPA violates the Establishment Clause, see Cutter v. Wilkinson (2005).