The Kronecker product of two Schur functions arises naturally in the study of the representation theory of the symmetric group and has long been a central object in algebraic combinatorics. Despite its fundamental role in algebraic combinatorics, many basic questions about this product remain open. A conjecture of Monical–Tokcan–Yong proposes that the monomial expansion of any Kronecker product has a saturated Newton polytope, a property that would provide new structural insight into these coefficients. In this talk, we will dicuss special cases of this conjecture that we are able to prove. Our methods also lead to new necessary conditions for the positivity of Kronecker coefficients. This is joint work with Greta Panova.