Polynomial relations for $q$-characters via the ODE/IM correspondence

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 $U_q(\mathfrak{b})$.