On products of sl_n characters and support containment

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