  The orbits of the symplectic group acting on the type A flag variety are
indexed by the fixed-point-free involutions in a finite symmetric group. The
cohomology classes of the closures of these orbits have polynomial
representatives $\hat{\mathfrak{S}}^{\tt{FPF}}_z$ akin to Schubert polynomials.
We show that the fixed-point-free involution Stanley symmetric functions
$\hat{F}^{\tt{FPF}}_z$, which are stable limits of the polynomials
$\hat{\mathfrak{S}}^{\tt{FPF}}_z$, are Schur $P$-positive. To do so, we
construct an analogue of the Lascoux-Sch\"utzenberger tree, an algebraic
recurrence that computes Schubert polynomials. As a byproduct of our proof, we
obtain a Pfaffian formula of geometric interest for
$\hat{\mathfrak{S}}^{\tt{FPF}}_z$ when $z$ is a fixed-point-free version of a
Grassmannian permutation. We also classify the fixed-point-free involution
Stanley symmetric functions that are single Schur $P$-functions, and show that
the decomposition of $\hat{F}^{\tt{FPF}}_z$ into Schur $P$-functions is
unitriangular with respect to dominance order on strict partitions. These
results and proofs mirror previous work by the authors related to the
orthogonal group action on the type A flag variety.
