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Diagram automorphisms and canonical bases for quantum affine algebras, II

Let ${\mathbf U}_q^-$ be the negative part of the quantum enveloping algebra, and $σ$ the algebra automorphism on ${\mathbf U}_q^-$ induced from a diagram automorphism. Let $\underline{\mathbf U}_q^-$ be the quantum algebra obtained from $σ$, and $\widetilde{\mathbf B}$ (resp. $\widetilde{\underline{\mathbf B}}$) the canonical signed basis of ${\mathbf U}_q^-$ (resp. $\underline{\mathbf U}_q^-$). Assume that ${\mathbf U}_q^-$ is simply-laced of finite or affine type. In our previous papers [SZ1, 2], we have proved by an elementary method, that there exists a natural bijection $\widetilde{\mathbf B}^σ \simeq \widetilde{\underline{\mathbf B}}$ in the case where $σ$ is admissible. In this paper, we show that such a bijection exists even if $σ$ is not admissible, possibly except some small rank cases.