  We prove the Kazhdan-Lusztig correspondence for a class of vertex operator
superalgebras which, via the work of Costello-Gaiotto, arise as boundary VOAs
of topological B twist of 3d $\mathcal{N}=4$ abelian gauge theories. This means
that we show equivalences of braided tensor categories of modules of certain
affine vertex superalgebras and corresponding quantum supergroups.
  We build on the work of Creutzig-Lentner-Rupert to this large class of VOAs
and extend it since in our case the categories don't have projective objectives
and objects can have arbitrary Jordan H\"older length.
  Our correspondence significantly improves the understanding of the braided
tensor category of line defects associated to this class of TQFT, by realizing
line defects as modules of a Hopf algebra. In the process, we prove a case of
the conjecture of Semikhatov-Tipunin, relating logarithmic CFTs to Nichols
algebras of screening operators.
