Modal companions for the special extensions of Nelson's constructive logic

The Belnapian version $\mathsf{BS4}$ of the normal modal logic $\mathsf{S4}$ is related to Nelson's constructive logic $\mathsf{N4}^{\bot}$ in approximately the same way as the logic $\mathsf{S4}$ is related to the intuitionistic logic. For this reason, it is natural to define modal companions for logics extending $\mathsf{N4}^{\bot}$ as extensions of the Belnapian modal logic $\mathsf{BS4}$. It is proved that, for every special extension $L$ of $\mathsf{N4}^{\bot}$, the logic $\tau^BL$, where $\tau^B$ is a natural modification of the mapping $\tau$ assigning the least modal companion to each superintuitionistic logic, is the least modal companion of $L$ in the lattice of extensions of $\mathsf{BS4}$.