Classify Lie superalgebra pairs with shifted induction–coinduction

Classify all pairs of Lie superalgebras for which the induction and coinduction functors differ only by a shift of the gradings and homological degree, and determine the size of this class.

Background

The paper proves that, for the principal parabolic inclusion used to categorify the genus-two Macdonald operators, the derived induction and coinduction functors are related by a grading shift and a homological shift. It notes that the same phenomenon occurs for inclusions of the form $$\mathfrak b\otimes\Bbbk[\xi]\subset\mathfrak p\otimes\Bbbk[\xi]$$, where $\mathfrak b$ and $\mathfrak p$ are a Borel and minimal parabolic subalgebra of a semisimple Lie algebra and $\xi$ is odd.