The oriented link of the cyclic quotient singularity Χ_(p, q) is orientation-preserving diffeomorphic to the lens space L(p, q) and carries the standard contact structure ζst. Lisca classified the Stein fillings of (L(p, q), ζst) up to diffeomorphisms and conjectured that they correspond bijectively through an explicit map to the Milnor fibres associated with the irreducible components (all of them being smoothing components) of the reduced miniversal space of deformations of Χ_(p, q). We prove this conjecture using the smoothing equations given by Christophersen and Stevens. Moreover, based on a different description of the Milnor fibres given by de Jong and van Straten, we also canonically identify these fibres with Lisca's fillings. Using these and a newly introduced additional structure (the order) associated with lens spaces, we prove that the above Milnor fibres are pairwise non-diffeomorphic (by diffeomorphisms which preserve the orientation and order). This also implies that de Jong and van Straten parametrize in the same way the components of the reduced miniversal space of deformations as Christophersen and Stevens.
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