Dual imaginary vectors, tight monomial cones and quantum Frobenius morphism

We show that the quantum Frobenius morphism and its splitting are not fully compatible with the canonical basis for any finite-dimensional simple Lie algebra if the rank is sufficiently large. The incompatibility occurs at same place where Leclerc found his imaginary vectors, and where there are monomials in the tight monomial cone which do not belong to the canonical basis.