Aims. To validate the Einstein equivalence principle (local position invariance) by limiting the fractional changes in the electron-to-proton mass ratio, mu = m_e/m_p, measured in Galactic plane objects. Methods. High resolution spectral observations of dark clouds in the inversion line of NH3(1,1) and pure rotational lines of other molecules (the so-called ammonia method) were performed at the Medicina 32-m and the Effelsberg 100-m radio telescopes to measure the radial velocity offsets, Delta RV = V_rot - V_inv, between the rotational and inversion transitions which have different sensitivities to the value of mu. Results. In our previous observations (2008-2010), a mean offset of <Delta RV> = 0.027+/-0.010 km/s [3 sigma confidence level (C.L.)] was measured. To test for possible hidden errors, we carried out additional observations of a sample of molecular cores in 2010-2013. As a result, a systematic error in the radial velocities of an amplitude ~0.02 km/s was revealed. The averaged offset between the radial velocities of the rotational transitions of HC3N(2-1), HC5N(9-8), HC7N(16-15), HC7N(21-20), and HC7N(23-22), and the inversion transition of NH3(1,1) <Delta RV> = 0.003+/-0.018 km/s (3 sigma C.L.). This value, when interpreted in terms of Delta mu/mu= (mu_obs - mu_lab)/mu_lab, constraints the mu-variation at the level of Delta mu/mu < 2*10^{-8} (3 sigma C.L.), which is the most stringent limit on the fractional changes in mu based on astronomical observations.
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