Q$_2$ sub-band, $\Delta \lambda$, as a function of $n_{2d}$, assuming $\omega_{ln}=230\pm30$K and $\mu^*=0.13$[@GePRB2013]. Filled blue circles are our experiments. Black and magenta open circles from Refs.. The blue dashed line is a guide to the eye.[]{data-label="figure:coupling"}](Figure_EPC_4.eps){width="\columnwidth"}

Fig.\[figure:coupling\]a shows that, for $0.23$% tensile strain, the experimentally observed positions of the kinks agree well with a linear extrapolation of the data of Ref. to 4L-MoS$_2$ (representative of our experiments) and for in-plane strain between $0\%$ (bulk) and $1.28\%$ (fully relaxed). These findings indicate that, while the mechanism proposed in Ref. qualitatively describes the general behavior of gated FL-MoS$_2$, quantitative differences arise due to the spin-orbit split of the Q$_1$ and Q$_2$ sub-bands. The main reason for the EPC (and, hence, $T_c$) increase is the same, i.e. the increase in the number of phonon branches involved in the coupling when the high-energy valleys are populated[@GePRB2013]. However, the finite spin-orbit-split between the sub-bands significantly alters the FS connectivity upon increasing doping[@BrummePRB2015]. If we consider the relevant phonon wave vectors ($q$=$\Gamma$,K,M,$\Gamma$K$/2$) for 1L- and FL-MoS$_2$[@MolinaPRB2011; @AtacaJPCC2011], and only the K/K’ valleys populated, then only phonons near $\Gamma$ and K can contribute to EPC[@GePRB2013]. The former strongly couple e$^-$ within the same valley[@GePRB2013], but cannot contribute 