Strongly fillable contact 3-manifolds without Stein fillings

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We use the Ozsvath-Szabo contact invariant to produce examples of strongly symplectically fillable contact 3-manifolds which are not Stein fillable.

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Tight contact structures on some small Seifert fibered 3-manifolds (2005, background): “(For further results on contact structures on these 3–manifolds see [ 7 , 39].) On the other hand, we are not aware of such an example where the underlying 3–manifold M is an L–space, that is a rational homology 3–sphere whose Heegaard Floer homology group b”
A survey of Heegaard Floer homology (2013, methodology): “However, the converse is not true, as shown by Ghiggini [20, 21]: there are tight contact structures with cpξq “ 0.”
Strongly fillable contact manifolds and $J$-holomorphic foliations (2008, background): “Remark 2.1. It was not known until recently whether strong and Stein fillability are equivalent notions: a negative answer was provided by a construction due to P. Ghiggini [ Ghi05 ] of strongly fillable contact manifolds that are not Stein fillable.”
Cabling, contact structures and mapping class monoids (2010, background): “The second inclusion is well known and the fact that the inclusion is strict follows from [26].”