  Schubert polynomials are refined by the key polynomials of
Lascoux-Sch\"{u}tzenberger, which in turn are refined by the fundamental slide
polynomials of Assaf-Searles. In this paper we determine which fundamental
slide polynomial refinements of key polynomials, indexed by strong
compositions, are multiplicity free. We also give a recursive algorithm to
determine all terms in the fundamental slide polynomial refinement of a key
polynomial indexed by a strong composition. From here, we apply our results to
begin to classify which fundamental slide polynomial refinements, indexed by
weak compositions, are multiplicity free. We completely resolve the cases when
the weak composition has at most two nonzero parts or the sum has at most two
nonzero terms.
