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We prove that flag versions of quiver Grassmannians (also knows as Lusztig'sfibers) for Dynkin quivers (types $A$, $D$, $E$) have no odd cohomology over$\mathbb{Z}$. Moreover, for types $A$ and $D$ we prove that these varietieshave $\alpha$-partitions into affine spaces. We also show that to prove thesame statement for type $E$, it is enough to check this for indecomposablerepresentations.  We also give a flag version of the result of Irelli-Esposito-Franzen-Reinekeon rigid representations: we prove that flag versions of quiver Grassmanniansfor rigid representations have a diagonal decomposition. In particular, theyhave no odd cohomology over $\mathbb{Z}$.

Flag versions of quiver Grassmannians for Dynkin quivers have no odd cohomology over $\mathbb{Z}$