The first detection of gravitational waves from the binary neutron star merger event GW170817 has started to provide important new constraints on the nuclear equation of state at high density. The tidal deformability bound of GW170817 combined with the observed two solar mass neutron star poses a serious challenge to theoretical formulations of realistic equations of state. We analyze a fully comprehensive set of relativistic nuclear mean-field theories by confronting them with the observational bounds and the measured neutron-skin thickness. We find that only a few models can withstand these bounds which predict a stiff overall equation of state but with a soft neutron-proton symmetry energy. Two possible indications are proposed: Circumstantial evidence of hadron-quark phase transition inside the star and new parametrizations that are consistent with ground state properties of finite nuclei and observational bounds. Based on extensive analysis of these sets, an upper limit on the radius of a $1.4M_odot$ neutron star of $R_{1.4}lesssim 12.9$ km is deduced.
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