Abstract: Let (M,g) be a four-dimensional closed connected oriented (possibly non-spin) Riemannian manifold with scalar curvature bounded below by 12. We prove that, if f is a smooth distance non-increasing map of non-zero degree from (M, g) to the unit four-sphere, then f is an isometry. This removes the spin condition in Llarull's scalar curvature rigidity theorem of spheres in dimension four. We utilize mu-bubbles and a version with coefficients of the rigidity of the three-sphere to rule out the case where all the inequalities are strict. Our proof of rigidity exploits monotonicity results for the harmonic map heat flow coupled with the Ricci flow due to Lee and Tam This is joint work with J. Wang, Z. Xie and B. Zhu.