How can a change in molecular structure affect the relative stability and structural properties of the twist-bend nematic phase ()? Here we extend the mean-field model (C. Greco, G. R. Luckhurst and A. Ferrarini, Soft Matter, 2014, 10, 9318) for bent-shaped achiral molecules, to study the influence of arm molecular biaxiality and the value of the molecule's bend angle on the relative stability of . In particular we show that by controlling the biaxiality of the molecule's arms, up to four ordered phases can become stable. They involve local uniaxial and biaxial variants of , together with uniaxial and biaxial nematic phases. However, a V-shaped molecule shows a stronger ability to form stable than a biaxial nematic phase, where the latter phase appears in the phase diagram only for bend angles greater than and for large biaxiality of the two arms.