Linearly Independent Products of Rectangularly Complementary Schur Functions

Fix a rectangular Young diagram R, and consider all the products of Schur functions s(mu) s(mu^c), where mu and mu^c run over all (unordered) pairs of partitions which are complementary with respect to R. Theorem: The self-complementary products, s(mu)^2 where mu=mu^c, are linearly independent of all other s(mu) s(mu^c). Conjecture: The products s(mu) s(mu^c) are all linearly independent.