We report on the current status of our on-going project to develop well-behaved hypersequent calculi for intermediate predicate logics, such as the linearity axiom $\mathbf{LIN}\colon\left(φ\toψ\right)\lor\left(ψ\toφ\right)$ and the constant domain axiom $\mathbf{CD}\colon\forall x\left(φ\lorψ\left(x\right)\right)\toφ\lor\forall xψ\left(x\right)$.