text: To address this issue, we took a larger λ for the same 
  ωsf and re-did the calculation of the conductivities and 
  optical integrals. The results for σ(ω) and ∆W (ωc) are 
  presented in Fig. 22. We found the same behavior as be- 
  fore, i.e., ∆WK is negative. But we also found that the 
  larger is the overall scale for the self-energy, the larger is a 
  frequency of zero-crossing of ∆W (ωc). In particular, for 
  the same λ and ωsf that were used in Ref. 33 to ﬁt the NS 
  conductivity data, the zero crossing is at ∼ 0.8 eV which 
  is quite close to the bandwidth. This implies that at a 
  truly strong coupling the frequency at which ∆W (ωc) 
  changes sign can well be larger than the bandwidth of 
  1eV in which case ∆W integrated up to the bandwidth 
  does indeed remain positive. Such behavior would be 
  consistent with Refs.8,9. we also see from Fig. 22 that 
  ∆WK becomes small at a truly strong coupling, and over 
  a wide range of frequencies the behavior of ∆W (ωc) is 
  predominantly governed by ∆f (ωc), i.e. by the cut-oﬀ 
  term.50 The implication is that, to ﬁrst approximation, 
  ∆WK can be neglected and positive ∆W (wc) integrated 
  to a frequency where it is still positive is almost compen- 
  sated by the integral over larger frequencies. This again 
  would be consistent with the experimental data in Refs. 
  8,9. 