On the gluing of hyperconvex metrics and diversities

In this work we consider two hyperconvex diversities (or hyperconvex metric spaces) (X, δ_X) and (Y, δ_Y ) with nonempty intersection and we wonder whether there is a natural way to glue them so that the new glued diversity (or metric space) remains being hyperconvex. We provide positive and negative answers in both situations.