  In Schubert Puzzles and Integrability I we proved several "puzzle rules" for
computing products of Schubert classes in K-theory (and sometimes equivariant
K-theory) of d-step flag varieties. The principal tool was "quantum
integrability", in several variants of the Yang--Baxter equation; this let us
recognize the Schubert structure constants as q->0 limits of certain matrix
entries in products of R- (and other) matrices of quantized affine algebra
representations. In the present work we give direct cohomological
interpretations of those same matrix entries but at finite q: they compute
products of "motivic Segre classes", closely related to K-theoretic
Maulik--Okounkov stable classes living on the cotangent bundles of the flag
varieties. Without q->0, we avoid some divergences that blocked fuller
understanding of d=3,4. The puzzle computations are then explained (in
cohomology onlyin this work, not K-theory) in terms of Lagrangian convolutions
between Nakajima quiver varieties. More specifically, the conormal bundle to
the diagonal inclusion of a flag variety factors through a quiver variety that
is not a cotangent bundle, and it is on that intermediate quiver variety that
the R-matrix calculation occurs.
