This technical note addresses the structure preserving stabilization by output feedback of conservative input-output contact systems, a class of input-output Hamiltonian systems defined on contact manifolds. In the first instance, achievable contact forms in closed-loop and the associated Legendre submanifolds are analysed. In the second instance the stability properties of a hyperbolic equilibrium point of a strict contact vector field are analysed and it is shown that the stable and unstable manifolds are Legendre submanifolds. In the third instance the consequences for the design of stable structure preserving output feedback are derived: in closed-loop one may achieve stability only relatively to some invariant Legendre submanifold of the closed-loop contact form and furthermore this Legendre submanifold may be used as a control design parameter. The results are illustrated along the technical note on the example of heat transfer between two compartments and a controlled thermostat.