significantly due to the limited size of the Fermi sheets[@GePRB2013]. The latter couple e$^-$ across different valleys[@GePRB2013], and provide a larger contribution[@GePRB2013], insufficient to induce SC. MoS$_2$ flakes are metallic but not superconducting before the crossing of Q$_1$. When this crossing happens, the total EPC increases due to the contribution of longitudinal phonon modes near K[@GePRB2013] (coupling states near two different Q or Q’), near $\Gamma$K$/2$[@GePRB2013] (coupling states near Q to states near Q’), and near M[@GePRB2013] (coupling states near Q or Q’ to states near K or K’). However, this first EPC increase associated with Q$_1$ is not sufficient to induce SC, as the SC transition is not observed until immediately before the crossing of the spin-orbit-split sub-band Q$_2$ and the second doping-induced Lifshitz transition. Additionally, the SC dome shows a maximum in the increase of $T_c$ with doping ($dT_c/dn_{2d}$) across the Q$_2$ crossing, i.e. when a new FS emerges. Consistently, the subsequent reduction of $T_c$ for $n_{2d}\geq 13\cdot 10^{13}$cm$^{-2}$ can be associated with the FS shrinkage and disappearance at K/K’[@GePRB2013; @BrummePRB2015], and might also be promoted by the formation of an incipient Charge Density Wave[@RosnerPRB2014; @Piattiarxiv2018] (characterized by periodic modulations of the carrier density coupled to a distortion 