Dipper-James-Murphy's conjecture for Hecke algebras of type B

We prove a conjecture by Dipper, James and Murphy that a bipartition is restricted if and only if it is Kleshchev. Hence the restricted bipartitions naturally label the crystal graphs of level two irreducible integrable $\mathcal{U}_v({\hat{\mathfrak{sl}}_e})$-modules and the simple modules of Hecke algebras of type $B_n$.
Comment: The revised version corrects minor points, the proof of lemma 3.3 has been improved