7 . 0 . 0 . 0 . 0 e e r i m t o t a r a t u r e ( K h 2 0 4 0 0 . 0 . 1 0 . 2 . 3 GLYPH<s61649> a x S F GLYPH<s61649> ( m m . 0 . 0 . 0 . 0 e e r i m t o t a r a t u r e ( K h 2 0 4 0 0 . 0 . 1 0 . 2 . 3 GLYPH<s61649> ( m GLYPH<s61649> a x S F m FIG. S4: The best fit to the free-energy difference of the four-band Eliashberg model with the coupling function B (Ω) peaked at $_{ℏ}$Ω max SF = 18 meV ( left ) and 13 meV ( right ) and parameters listed in Table SII.. The red, green, and blue lines represent the partial contributions of the individual bands. . 0 . 0 . 0 r a t u r e ( K GLYPH<s61649> a x S F m . 0 . 0 . 0 r a t u r e ( K GLYPH<s61649> a x S F m FIG. S5: Temperature dependence of the superconducting energy gaps calculated from the four-band Eliashberg model with the coupling function B (Ω) peaked at $_{ℏ}$Ω max SF = 18 meV ( left ) and 13 meV ( right ) and parameters listed in Table SII. 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 . [S2] J.W. Loram et al ., Physica C 168 , 47 (1990). [S1] J.W. Loram, J. Phys. C 19 , 6113 (1986). [S3] Y. Wang et al. , Physica C 355 , 179 (2001). [S4] D. Reznik et al. , Phys. Rev. B 80 , 214534 (2009). [S5] R. J. Radtke et al. , Phys. Rev. B 46 , 11975 (1992). [S6] A. Millis, H. Monien, D. Pines, Phys. Rev. B 42 , 167 (1990). [S7] H.-B. Sch¨uttler, M.R. Norman, Phys. Rev. B 54 , 13295 (1996). [S8] L. Benfatto et al., Phys. Rev. B 80 , 214522 (2009). [S9] D. S. Inosov et al., Nature Phys. 6 , 178 (2010) [S10] C. Lester et al., Phys. Rev. B 81 , 064505 (2010). [S11] A. N. Yaresko, private communication.