With archival and new Hubble Space Telescope observations we have refined the space-velocity measurements of the stars in the central region of the remnant of Tychos supernova (SN) 1572, one of the historical Galactic Type Ia supernova remnants (SNRs). We derived a proper motion for Tycho-G of (mu_RA_cos_dec;mu_dec)=(-2.63;-3.98)+/-(0.06;0.04)[formal errors]+/-(0.18;0.10)[expected errors] mas/yr. We also reconstruct the binary orbit that Tycho-G should have followed if it were the surviving companion of SN 1572. We redetermine the Ni abundance of this star and compare it with new abundance data from stars of the Galactic disk, finding that [Ni/Fe] is about 1.7 sigma above the Galactic trend. From the high velocity (v_b = -50+/-14 km/s) of Tycho-G perpendicular to the Galactic plane, its metallicity, and its Ni excess, we find the probability of its being a chance interloper to be P < 0.00037 at most. The projected rotational velocity of the star should be below current observational limits. The projected position of Tycho-G is, within the uncertainties, consistent with the centroid of the X-ray emission of Tychos SNR; moreover, its brightness is generally consistent with the post-explosion evolution of the luminosity of a SN companion. Among the other 23 stars having V<22 mag and located within 42 arcsec from the X-ray centroid, only 4 are at distances compatible with that of the SNR, and none of them shows any peculiarity. Therefore, if even Tycho-G is not the surviving companion of SN 1572, the absence of other viable candidates does favor the merging of two white dwarfs as the producer of the SN.
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