  The wide binary orbit of Proxima Centauri around $\alpha$ Centauri A and B
differs significantly between Newtonian and Milgromian dynamics (MOND). By
combining previous calculations of this effect with mock observations generated
using a Monte Carlo procedure, we show that this prediction can be tested using
high precision astrometry of Proxima Centauri. This requires ${\approx 10}$
years of observations at an individual epoch precision of $0.5 \, \mu$as,
within the design specifications of the proposed Theia mission. In general, the
required duration should scale as the 2/5 power of the astrometric precision. A
long-period planet could produce a MOND-like astrometric signal, but only if it
has a particular ratio of mass to separation squared and a sky position close
to the line segment connecting Proxima Centauri with $\alpha$ Centauri.
Uncertainties in perspective effects should be small enough for this test if
the absolute radial velocity of Proxima Centauri can be measured to within
${\approx 10}$ m/s, better than the present accuracy of 32 m/s. We expect the
required improvement to become feasible using radial velocity zero points
estimated from larger samples of close binaries, with the Sun providing an
anchor. We demonstrate that possible astrometric microlensing of Proxima
Centauri is unlikely to affect the results. We also discuss why it should be
possible to find sufficiently astrometrically stable reference stars.
Adequately addressing these and other issues would enable a decisive test of
gravity in the currently little explored low acceleration regime relevant to
the dynamical discrepancies in galactic outskirts.
