this approach, $\mathbb{T}^{\bf b}$ (more precisely, its integral form) at $q = 1$ is identified with the Grothendieck group $[\mc{O}^{\bf b}]$ of the BGG category $\mc{O}^{\bf b}$ of $\mathfrak{osp}(2m|2n)$-modules (relative to a Borel subalgebra of type ${\bf b}$). We construct the $\imath$-canonical basis and dual $\imath$-canonical basis on (a suitable completion of) the tensor space $\mathbb{T}^{\bf b}$. The construction of these bases is exactly the same as in [@BW13], while only the precise formulas of these bases are different (which is irrelevant to the construction).

For the Lie superalgebra $\mathfrak{osp}(2m|2n)$, there are generally two types of fundamental systems, hence related Dynkin diagrams, with respect to different choices of the Borel subalgebras ${\bf b}$: with a type D branch (where ${\bf b}$ starts with $0^2$) in the Dynkin diagram; or with a type C branch (where ${\bf b}$ starts with $1$) in the Dynkin diagram. Those two types of fundamental systems are not conjugate under the Weyl group actions, but differ by odd reflections. We prove the Kazhdan-Lusztig theory for the category $\mc{O}^{\bf b}$ of the Lie superalgebra $\mathfrak{osp}(2m|2n)$ with respect to ${\bf b}$ of the first type by induction on $n$, where the base case $n=0$ follows from the classical 