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

Character formulae for queer Lie superalgebras and canonical bases of types $A/C$