Affine $\mathrm{i}$quantum groups and Steinberg varieties of type C, II

A geometric realization of the quasi-split affine iquantum group of type ${\mathrm{A}\mathrm{I}\mathrm{I}\mathrm{I}}_{2n-1}^{(\tau )}$ was given by Wang and the second author, in terms of equivariant K-groups of Steinberg varieties of type C. As a completion of that work, this paper focuses on the previously untreated case. We provide a similar construction of the quasi-split affine iquantum group of type ${\mathrm{A}\mathrm{I}\mathrm{I}\mathrm{I}}_{2n}^{(\tau )}$, using the same equivariant K-groups of Steinberg varieties of type C. In the appendix, we employ Steinberg varieties of type D to give a new realization of the quasi-split affine iquantum group of type ${\mathrm{A}\mathrm{I}\mathrm{I}\mathrm{I}}_{2n-1}^{(\tau )}$, thereby avoiding the localization method adopted in the previous work.