Pontryagin algebras and the LS-category of moment-angle complexes in the flag case

We study the Pontryagin algebras (the loop homology) of moment-angle complexes corresponding to flag simplicial complexes. The Poincare series, an explicit minimal set of generators and full description of free and one-relator such algebras are known due to the works of Panov-Ray, Grbic-Panov-Theriault-Wu and Grbic-Ilyasova-Panov-Simmons.
We introduce the natural multigrading and describe the degrees of minimal relations and multigraded Poincare series. We prove the collapse of the Milnor-Moore spectral sequence, and use this to calculate the Lusternik-Schnirelmann category of such moment-angle complexes.