We have shown previously that reducing sufficiently the value of the weak force Fermi constant, GF, leads to a helium dominated universe. The reason is that the smaller GF, the earlier the freeze-out of the leptonic reactions, and the higher the temperature. Hence the closer to equality is the density of neutrons and protons, and an all-helium universe results.
However, we have taken for granted that the neutrons and protons achieve their thermal equilibrium densities. This will only be the case if the weak interactions are active, since this provides the mechanism for their inter-conversion. Thus, we have an entirely different situation if the weak interaction does not exist at all. In this case, the relative abundance of protons and neutrons would be determined by whatever CP symmetry violating mechanism gives rise to baryogenesis. In other words, we can presumably fix the relative neutron and proton abundance as we wish. So there is no reason to assume equal numbers of protons and neutrons, and hence an all-helium universe can be avoided.
Harnik, Kribs and Perez have constructed a universe along these lines. The key features of this fascinating paper are described in this Chapter. Despite having no weak nuclear force it is shown that the HKP universe will give rise to stars and to heavy elements, and thus apparently support the emergence of life. This construction was specifically designed by HKP to refute the suggestion of an anthropic constraint of the weak force.
Admittedly, this would not apply in our universe, in the sense that other parameters are altered in addition to GF in order to make this 'weakless' universe work. This construction serves as a warning that definitive conclusions about fine-tuning cannot be drawn from varying single parameters alone.
