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We add four warmer late-type supergiants to our previousspectro-interferometric studies of red giants and supergiants.  V766 Cen (=HR 5171 A) is found to be a high-luminosity log(L/L_sun)=5.8+-0.4source of Teff 4290+-760 K and radius 1490+-540 Rsun located close to both theHayashi and Eddington limits; this source is consistent with a 40 Msunevolutionary track without rotation and current mass 27-36 Msun. It exhibitsNaI in emission arising from a shell of radius 1.5 Rphot and a photocenterdisplacement of about 0.1 Rphot. V766 Cen shows strong extended molecular (CO)layers and a dusty circumstellar background component. This suggest anoptically thick pseudo-photosphere at about 1.5 Rphot at the onset of the wind.V766 Cen is a red supergiant located close to the Hayashi limit instead of ayellow hypergiant already evolving back toward warmer Teff as previouslydiscussed.  The stars sigma Oph, BM Sco, and HD 206859 are found to have lowerluminosities of about log(L/Lsun)=3.4-3.5 and Teff of 3900-5300 K,corresponding to 5-9 Msun tracks. They do not show extended molecular layers asobserved for higher luminosity red supergiants of our sample. BM Sco shows anunusually strong contribution by an over-resolved circumstellar dust component.These stars are more likely high-mass red giants instead of red supergiants.  This leaves us with an unsampled locus in the HR diagram corresponding toluminosities log(L/Lsun)~3.8-4.8 or masses 10-13 Msun, possibly correspondingto the mass region where stars explode as type II-P supernovae during the RSGstage.  Our previously found relation of increasing strength of extended molecularlayers with increasing luminosities is now confirmed to extend to double ourprevious luminosities and up to the Eddington limit. This might further pointto steadily increasing radiative winds with increasing luminosity.  [Abridged]

VLTI/AMBER spectro-interferometry of the late-type supergiants V766 Cen (=HR 5171 A), sigma Oph, BM Sco, and HD 206859