  We prove sign-alternation of the structure constants in the basis of
structure sheaves of opposite Schubert varieties in the torus-equivariant
Grothendieck group of coherent sheaves on the flag varieties $G/P$ associated
to an arbitrary symmetrizable Kac-Moody group $G$, where $P$ is any parabolic
subgroup. This generalizes the work of Anderson-Griffeth-Miller from the finite
case to the general Kac-Moody case, and affirmatively answers a conjecture of
Lam-Schilling-Shimozono regarding the signs of the structure constants in the
case of the affine Grassmannian.
