Do the current astronomical observations exclude the existence of non-strange quark stars?

As is pointed out in a recent work (Phys. Rev. Lett. 120, 222001), quark matter may not be strange. Inspired by this argument, we use a new self-consistent mean field approximation method to study the equation of state of cold dense matter within the framework of the two-flavor NJL model. Then the mass-radius relationship of two-flavor pure quark stars is studied. In the new self-consistent mean field approximation method we introduced, there is a free parameter $\alpha$, which reflects the weight of different interaction channels. In principal, $\alpha$ should be determined by experiments rather than the self-consistent mean field approximation itself. In this paper, thus, the influence of the variation of $\alpha$ on the critical chemical potential of chiral phase transition and the stiffness of the EOS are thoroughly explored. It is found that the stiffness of EOS can be increased greatly to support a two-solar-mass pure quark star when $\alpha$ is greater than 0.95. This means that current theoretical calculations and astronomical observations cannot rule out the possibility of a two-flavor pure quark star.