increase due to the appearance of e$^-$ pockets at Q$_2$. By setting $\omega_{ln}=230\pm30$K and $\mu^*=0.13$ (as for Ref.), and using Eq.\[EqAllenDynes\], we find that the limit of $\lambda(T_c)$ for $T_c\to0$ is $\sim0.25$. The corresponding $\Delta\lambda$ vs. $n_{2d}$ dependence is shown in Fig.\[figure:coupling\]b. The crossing at Q$_2$ results in a maximum $\Delta\lambda=0.63\pm0.1$, with a maximum EPC enhancement of $350\pm40\%$ with respect to the non-superconducting region. This indicates that the largest contribution to the total EPC, hence to the maximum $T_c\sim11$K, is associated with the population of the Q$_2$ sub-band. This is consistent with the reports of a reduced $T_c\sim2$K in 1L-MoS$_2$[@CostanzoNatNano2016; @FuQuantMater2017], shown to be superconducting for smaller $n_{2d}\sim5.5\cdot10^{13}$cm$^{-2}$[@FuQuantMater2017], hence likely to populate Q$_1$ only. $n_{2d}\sim5\cdot10^{13}$cm$^{-2}$ is also the doping expected for the crossing of Q$_1$ in 1L-MoS$_2$ in presence of a low-T strain similar to that in our 4L-MoS$_2$ devices (see Fig.\[figure:coupling\]a).

In summary, we exploited the large carrier density modulation provided by ionic gating to explore sub-band population and multivalley transport in MoS$_2$ layers. We detected two kinks in the conductivity, associated with the doping-induced crossing of the two sub-bands at Q/Q’. By comparing the emergence of these kinks with the doping dependence of $T_c$, we showed how superconductivity 