  In 2015, Jing and Li defined type B quasisymmetric Schur functions and
conjectured that these functions have a positive, integral and unitriangular
expansion into peak functions. We prove this conjecture, and refine their
combinatorial model to give explicit expansions in monomial, fundamental and
peak bases. We also show that these functions are not quasisymmetric Schur,
Young quasisymmetric Schur or dual immaculate positive, and do not have a
positive multiplication rule.
