In: Kazhdan-Lustzig polynomials and character formulae for the Lie superalgebra $gl(m|n)$, J. Amer. Math. Soc. 16 (2003), 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, Duke Math. J. Volume 164, Number 4 (2015), 617-695 and in:
Tensor Product Categorifications and the Super Kazhdan–Lusztig Conjecture, , IMRN, Volume 2017, Issue 20, 1 October 2017, Pages 6329–6410, independent proofs are provided.
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.