Character formulae for queer Lie superalgebras and canonical bases of types $A\mathrm{/}C$

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.