# Homology of the free loop space of generalized flag varieties

Is it known whether for a generalized complex flag variety $$X$$ (that is, $$G/P$$ for a complex semisimple Lie group $$G$$ and a parabolic $$P$$), the homology of the free loop space $$H_*(\Lambda X, \mathbb{Q})$$ is degree-wise finite-dimensional? Is it known at least for type A (i.e. classical) flag varieties?

This is known to be true for some concrete examples, such as complex projective spaces (Ziller et al.), and complete flag varieties of rank 2 (Burfitt-Grbić), but I was unable to find any statements for the general case.

Serre proved that for any simply-connected $$X$$, if $$X$$ has finitely generated homology groups in each degree, then the loop space of $$X$$ has finitely generated homology groups in each degree. (Proposition 9 of chapter IV of Homologie singulière des espaces fibrés. Applications. Ann. of Math., 54, 1951, p. 425-505.)