Ref. ($\sim8.6\pm4.1\mu$F/cm$^2$), where DEME-TFSI is the N,N-Diethyl-N-methyl-N-(2-methoxyethyl)ammonium bis(trifluoromethanesulfonyl)imide ionic liquid[@ShiSciRep2015].

Fig.\[figure:transport\]a shows that, while for T$\gtrsim$100K $R_s$ is a monotonically decreasing function of $n_{2d}$, the same does not hold for T$\lesssim$100K, where the various curves cross. In particular, the residual $R_s$ in the normal state $R_s^0$ (measured just above $T_c$ when the flake is superconducting) varies non-monotonically as a function of $n_{2d}$. This implies the existence of multiple local maxima in the $R_s^0 (n_{2d})$ curve. Consistently with the theoretical predictions of Ref., we find two local maxima. The first and more pronounced occurs when the flake is superconducting, i.e. for $n_{2d}>6\cdot 10^{13}$cm$^{-2}$. This feature was also reported in Refs., but not discussed. The second, less pronounced kink, is observed for $1\cdot 10^{13} \lesssim n_{2d} \lesssim 2\cdot 10^{13}$cm$^{-2}$, not previously shown. Both kinks can be seen only for T$\lesssim$70K and they are smeared for T$\gtrsim$150K.

The kink that emerges in the same range of $n_{2d}$ as the superconducting dome extends across a wide range of $V_{LG}$ ($3 \lesssim V_{LG} \lesssim 6$V) for $n_{2d}\gtrsim6\cdot10^{13}$cm$^{-2}$, and can be accessed only by LG biasing, due to the small capacitance of the solid BG. This prevents a continuous characterization of its behavior, as $n_{2d}$ induced 