In: [Kazhdan-Lustzig polynomials and character formulae for the Lie superalgebra $gl(m|n)$][1], J. Amer. Math. Soc. 16 (2002), 185–231, J. Brundan develops a conjecture on the characters for the irreducible modules and tilting modules in the full
BGG category $\mathcal{O}^{m|n}$ of $gl(m|n)$ modules. 

In: [The Brundan–Kazhdan–Lusztig conjecture for general linear Lie superalgebras][2], Duke Math. J. Volume 164, Number 4 (2015), 617-695 and in:  
[Tensor Product Categorifications and the Super Kazhdan–Lusztig Conjecture][3], , IMRN, Volume 2017, Issue 20, 1 October 2017, Pages 6329–6410, independent proofs are provided.   

(The arXiv versions are: [arXiv:1203.0092 \[math.RT\]][4] and [arXiv:1310.0349v3 \[math.RT\]][5] correspondingly). 

I am not a specialist to say more, but i think the results in these papers may be related to what you are looking for. 


  [1]: http://www.ams.org/journals/jams/2003-16-01/S0894-0347-02-00408-3/S0894-0347-02-00408-3.pdf
  [2]: https://projecteuclid.org/euclid.dmj/1426512104
  [3]: https://academic.oup.com/imrn/article/2017/20/6329/3061017
  [4]: https://arxiv.org/abs/1203.0092
  [5]: https://arxiv.org/abs/1310.0349