The Substrate influence on the measured STS gaps. As we have mentioned above, the main crucial property under possible dispute is the magnitude of the substrate influence (screening) ΔEsb on the measured STS gap. According to our earlier estimates ΔEsb should be of the order of a few 0.1 eV. However, almost in all cases ΔEsb larger than 1 eV is needed to bridge the experimental STS measurements for AGNRs deposited on metal surfaces (usually Au) and the theoretical values for free standing AGNRs. The theoretical values widely recognized as an almost universal point of reference are the GW results of Yang et al., which among the theoretical values reported earlier are clearly the largest, and many times by far. As a result, ΔEsb which is defined as the difference of the STS measurements and the theoretical reference values are unrealistically large. For example, for the 7-AGNRs the theoretical GW gap 14 is 3.7 eV, whereas the experimental STS gap value obtained by various groups 6-7, 9,15 is 2.5±0.2 eV. Thus, ΔEsb should be at least 1.2 eV. However, the STS value of 2.5 eV was also obtained for 7-AGNRs deposited on non-metallic substrates, such as NaCl 6 and MgO , for which such large ΔEsb value is clearly unrealistic. On the basis of their STS measurements on samples grown on MgO, Kolmer et al. concluded that ΔEsb should be marginal, which is in full agreement with our present results. However, the general consensus, with few exceptions is largely different (up to now). In this work we are led to conclude that the GW results overestimate the bandgaps mainly due to the size effect, since the GW results of Yang et al. were obtained for infinite AGNRs, whereas the atomically precise AGNRs have finite length (and topological end-states). This could be sufficient to explain the resulting unrealistically large ΔEsb values. Yet, besides the infinite size (and the corresponding periodic boundary conditions) the lack of exact exchange in the LGA wavefunctions building the Green's function could be also important since exchange interaction is very sensitive to inversion symmetry frustration. Nevertheless, judging from our TDDFT results, it is more reasonable to attribute the gap difference between the infinite and the finite size AGNRs (size effect) to the mixing of edge/end states with the infinite "bulk" states (and the scattering at the zigzag edges) which can drastically reduce the gap. This is corroborated by the GW results 6 of Wang et al. for the finite (7, 24) AGNR (and slightly longer), who obtained a gap of 2.8 eV clearly closer to the measured (by several groups) gap, and substantially smaller compared to the 3.7 eV (G0W0) value for the infinite 7-AGNR. Parenthetically, it should be mentioned at this point that even in the worst-case scenario where the substrate interaction is strong (especially when the distance of STS tip from the surface is small), this leads to mixed substrate-AGNR states which can be easily recognized (and excluded) from the measurements by comparing the two separate STS spectra. Moreover, such states would be expected to have low overlap with the pure AGNR-excited-states, and consequently the corresponding transition(s) would have very low intensity and would be difficult to detect. Thus, we assert here that ΔEsb should indeed be marginal, in full agreement with the experimental results (for 7-AGNRs grown directly on MgO substrate) and conclusions of Kolmer et al. .