I have been reading the book "Automorphic forms and Lie superalgebras". In Section 2.6, Definition 2.6.15 we have the definition of Category O for BKM Lie superalgebras (I have also checked the book infinite-dimensional Lie algebras by Minoru Wakimato). I am interested to know, whether this category (or integrable modules in this category) is completely reducible or not?
I know this is true for BKM algebras from this paper : Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara. Crystal bases for quantum generalized Kac-Moody algebras. Proc. Lond. Math. Soc. 90(3), no. 2, 395--438, 2005. but I couldn't guess or prove for the case of BKM superalgebras.
Kindly help me with this and suggest me some references. Thanking you.