We present transport measurements of cleaved edge overgrowth GaAs quantum wires. The conductance of the first mode reaches 2 e²/h at high temperatures T > 10 K, as expected. As T is lowered, the conductance is gradually reduced to 1 e²/h, becoming T-independent at T < 0.1 K, while the device cools far below 0.1 K. This behavior is seen in several wires, is independent of density, and not altered by moderate magnetic fields B. The conductance reduction by a factor of two suggests lifting of the electron spin degeneracy in absence of B. Our results are consistent with theoretical predictions for helical nuclear magnetism in the Luttinger liquid regime.