Schur-positivity in a Square

Determining if a symmetric function is Schur-positive is a prevalent and, in general, a notoriously difficult problem. In this paper we study the Schur-positivity of a family of symmetric functions. Given a partition \lambda, we denote by \lambda^c its complement in a square partition (m^m). We conjecture a Schur-positivity criterion for symmetric functions of the form s_{\mu'}s_{\mu^c}-s_{\lambda'}s_{\lambda^c}, where \lambda is a partition of weight |\mu|-1 contained in \mu and the complement of \mu is taken in the same square partition as the complement of \lambda. We prove the conjecture in many cases.