Conjectured decidability of several membership and consistency problems

Prove that the membership problems for Lbc, Ldc, Lbdc, Lfbdc, Lutra, and Ldtra and the consistency problems for Lbc, Ldc, Lbdc, Lfbdc, Lutra, and Ldtra are decidable.

Background

Beyond the base results achieved in the paper, the authors put forward a broader decidability conjecture for both membership and consistency problems across several intuitionistic modal systems defined by confluence and (up/down) transitivity conditions.