  We study the motivic Pandharipande-Thomas invariants of the Enriques
Calabi-Yau threefolds in fiber curve classes by basic computations and analysis
of a wallcrossing formula of Toda. Motivated by our results we conjecture a
formula for the perverse Hodge numbers of the compactified Jacobian fibration
of linar systems on Enriques surfaces in terms of its Betti numbers. This leads
to an asymptotic for said Hodge numbers and raises questions about the
behaviour of the extremal Hodge numbers.
