Does glycerinated skeletal muscle fibers need ATP in order to contract?

No.

One fundamental role of ATP in muscle contraction is to break the bonds between actin and myosin after they have binded together during the contractile process. This will occur spontaneously in glycerinated muscle fibers as the bathing solution contains relatively high levels of Mg²⁺. Mg²⁺ can bind to the myosin head and displace ADP from the myosin active site, releasing myosin from actin in a process known as "rigor."