On products of characters and support containment

Let λ, μ, ν and ρ be dominant weights of satisfying λ+μ=ν+ρ. Let Vλ denote the highest weight module corresponding to λ. Lam, Postnikov, Pylyavskyy conjectured a sufficient condition for Vλ Vμ to be contained in Vν Vρ as -modules. In this note we prove a weaker version of the conjecture. Namely we prove that under the conjectured conditions every irreducible -module which appears in the decomposition of Vλ Vμ does appear in the decomposition of Vν Vρ.