  Let $U_q(\mathfrak{b})$ be the Borel subalgebra of a quantum affine algebra
of type $X^{(1)}_n$ ($X=A,B,C,D$). Guided by the ODE/IM correspondence in
quantum integrable models, we propose conjectural polynomial relations among
the $q$-characters of certain representations of $U_q(\mathfrak{b})$.
