  The oriented link of the cyclic quotient singularity
  $\mathcal{X}_{p,q}$ is orientation-preserving diffeomorphic to the lens space
$L(p,q)$ and carries the standard contact structure $\xi_{st}$. Lisca
classified the Stein fillings of $(L(p,q), \xi_{st})$ up to diffeomorphisms and
conjectured that they correspond bijectively through an {\it explicit} map to
the Milnor fibers associated with the irreducible components (all of them being
smoothing components) of the reduced miniversal space of deformations of
$\mathcal{X}_{p,q}$. We prove this conjecture using the smoothing equations
given by Christophersen and Stevens. Moreover, based on a different description
of the Milnor fibers given by de Jong and van Straten, we also canonically
identify these fibers with Lisca's fillings. Using these and a newly introduced
additional structure - the order - associated with lens spaces, we prove that
the above Milnor fibers 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.
