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 Tycho's supernova (SN) 1572, one of the historical Galactic Type Ia supernova remnants (SNRs). We derived a proper motion for Tycho-G of (μαcos δ; μδ)J2000.0 = (-2.63; -3.98) ± (0.06; 0.04) [formal errors] ± (0.18; 0.10) [expected errors] mas yr-1. If the star were at the distance of the SNR (taken here to be 2.83 kpc), its velocity would be vb = -51 ± 1.5 km s-1. 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 disc, finding that [Ni/Fe] is about 1.7σ above the Galactic trend. From the high velocity of Tycho-G perpendicular to the Galactic plane ( - 90 ± 3 km s-1 ≤ vb ≤ -45 ± 1 km s-1, for the allowed range of distances to the star), its metallicity and its Ni excess, we find the probability of it being a chance interloper to be P ≲ 0.000 37 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 Tycho's SNR; moreover, its brightness is generally consistent with the post-explosion evolution of the luminosity of an 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 favour the merging of two white dwarfs as the producer of the SN.