A conjecture on monomial realizations and polyhedral realizations for crystal bases

Crystal bases are powerful combinatorial tools in the representation theory of quantum groups $U_q(\mathfrak{g})$ for a symmetrizable Kac-Moody algebras $\mathfrak{g}$. The polyhedral realizations are combinatorial descriptions of the crystal base $B(\infty)$ for Verma modules in terms of the set of integer points of a polyhedral cone, which equals the string cone when $\mathfrak{g}$ is finite dimensional simple. It is a fundamental and natural problem to find explicit forms of the polyhedral cone. The monomial realization expresses crystal bases $B(\lambda)$ of integrable highest weight representations as Laurent monomials with double indexed variables. In this paper, we give a conjecture between explicit forms of the polyhedral cones and monomial realizations. We prove the conjecture is true when $\mathfrak{g}$ is a classical Lie algebra, a rank $2$ Kac-Moody algebra or a classical affine Lie algebra.