  "Why is the EoS for tin so soft?" is a longstanding question, which prevents
us from determining the nuclear incompressibility $K_\infty$ accurately. To
solve this puzzle, a fully self-consistent quasiparticle random phase
approximation (QRPA) plus quasiparticle-vibration coupling (QPVC) approach
based on Skyrme-Hartree-Fock-Bogoliubov is developed. We show that the
many-body correlations introduced by QPVC, which shift the ISGMR energy in Sn
isotopes by about 0.4 MeV more than the energy in $^{208}$Pb, play a crucial
role in providing a unified description of the ISGMR in Sn and Pb isotopes. The
best description of the experimental strength functions is given by SV-K226 and
KDE0, which are characterized by incompressibility values $K_\infty=$ 226 MeV
and 229 MeV, respectively, at mean field level.
