We measured the ratio \(P_{x}/P_{z}\) of the transverse to longitudinal components of polarization transferred from electrons to bound protons in \(^{12}\mathrm{C}\) by the \(^{12}\mathrm{C}(\vec{e},e'\vec{p})\) process at the Mainz Microtron (MAMI). We observed consistent deviations from unity of this ratio normalized to the free-proton ratio, \((P_{x}/P_{z})_{^{12}\mathrm{C}}/(P_{x}/P_{z})_{^{1}\mathrm{H}}\), for both \(s\)- and \(p\)-shell knocked out protons, even though they are embedded in time averaged local densities that differ by about a factor of two. The dependence of the double ratio on proton virtuality is similar to the one for knocked out protons from \(^{2}\mathrm{H}\) and \(^{4}\mathrm{He}\), suggesting a universal behavior. It further implies no dependence on average local nuclear density.