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best simultaneous fit to the experimental values of T$_{c}$ , the superconducting gaps, and the temperature dependence of the free- energy difference. The effective coupling constant averaged over all bands, λ av = ∑ ij N h,e i (0) λ$_{ij}$/N h,e tot ≈ 1 . 9 , is remarkably close to the coupling constant of Pb$_{0}$$_{.}$$_{8}$Bi$_{0}$$_{.}$$_{2}$ and confirms our conclusion about strong coupling in Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ (see Fig.3 of the Letter for comparison). The consistency with the experiment tolerates some variation of the parameters and remains satisfactory for $_{ℏ}$Ω max SF within the 10 - 20 meV energy range. In Table SII and Figs. S4-S5 we compare our results for two representative coupling functions with $_{ℏ}$Ω max SF = 18 meV and 13 meV, illustrating that a shift of $_{ℏ}$Ω max SF to lower energy leads to a larger values of λ av with corresponding densities of states N h,e i (0) approaching the first-principle Density Functional results that accounts for the strong renormalization of the Sommerfeld constant γ N $_{K}$/γ$_{DFT}$ ∼ 5 . 

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