  This paper presents two existence h-principles, the first for conformal
symplectic structures on closed manifolds, and the second for leafwise
conformal symplectic structures on foliated manifolds with non empty boundary.
The latter h-principle allows to linearly deform certain codimension-$1$
foliations to contact structures. These results are essentially applications of
the Borman-Eliashberg-Murphy h-principle for overtwisted contact structures and
of the Eliashberg-Murphy symplectization of cobordisms, together with tools
pertaining to foliated Morse theory, which are elaborated here.
