that the incompressibility $K_{0}$ of symmetric nuclear matter can be extracted from a self-consistent microscopic theoretical model that successfully reproduces the experimental ISGMR energies as well as the ground state binding energies and charge radii of a variety of nuclei [Bla80]{}. Experimentally, thanks to new and improved experimental facilities and techniques, the ISGMR centroid energy $E_{\mathrm{ISGMR}}$, i.e., the so-called nuclear breathing mode energy, of $^{208}$Pb (a heavy, doubly-magic nucleus with a well-developed monopole peak) has been measured with a very high precision (less than $2\%$). Indeed, a value of $E_{\mathrm{ISGMR}}=14.17\pm 0.28$ MeV was extracted from the giant monopole resonance in $^{208}$Pb based on an improved $\alpha $-scattering experiment [You99]{} (another value of $E_{\mathrm{ISGMR}}=13.96\pm 0.20$ MeV was extracted in Ref. [@You04]). The $E_{\mathrm{ISGMR}}$ of $^{208}$Pb has been extensively used to constrain the $K_{0}$ parameter in the literature [@You99; @You04; @Lui04; @Ma02; @Vre03; @Col04; @Shl06; @LiT07; @Gar07; @Paa07; @Col09; @Pie10]. It is thus important to estimate and eventually narrow down the theoretical uncertainty of extracting $K_{0}$ from the nuclear ISGMR. Theoretically, in fact, it has been found that the uncertainty of the density dependence of the symmetry energy has significantly influenced the precise extraction of the $K_{0}$ parameter from ISGMR in $^{208}$Pb 