  The $K$-theoretic Schur $P$- and $Q$-functions $GP_\lambda$ and $GQ_\lambda$
may be concretely defined as weight generating functions for semistandard
shifted set-valued tableaux. These symmetric functions are the shifted
analogues of stable Grothendieck polynomials, and were introduced by Ikeda and
Naruse for applications in geometry. Nakagawa and Naruse specified families of
dual $K$-theoretic Schur $P$- and $Q$-functions $gp_\lambda$ and $gq_\lambda$
via a Cauchy identity involving $GP_\lambda$ and $GQ_\lambda$. They conjectured
that the dual power series are weight generating functions for certain shifted
plane partitions. We prove this conjecture. We also derive a related generating
function formula for the images of $gp_\lambda$ and $gq_\lambda$ under the
$\omega$ involution of the ring of symmetric functions. This confirms a
conjecture of Chiu and the second author. Using these results, we verify a
conjecture of Ikeda and Naruse that the $GQ$-functions are a basis for a ring.
