  For the BGG category of $\mathfrak{q}(n)$-modules of half-integer weights, a
Kazhdan-Lusztig conjecture \`a la Brundan is formulated in terms of categorical
canonical basis of the $n$th tensor power of the natural representation of the
quantum group of type $C$. For the BGG category of $\mathfrak{q}(n)$-modules of
congruent non-integral weights, a Kazhdan-Lusztig conjecture is formulated in
terms of canonical basis of a mixed tensor of the natural representation and
its dual of the quantum group of type $A$. We also establish a character
formula for the finite-dimensional irreducible $\mathfrak{q}(n)$-modules of
half-integer weights in terms of type $C$ canonical basis of the corresponding
$q$-wedge space.
