Distance to Proxima Centauri (Gaia VS New Horizons parallax program) The current best parallax measurement for the nearest star to the Solar System, Proxima Centauri, has been given by the European Space Agency's Gaia mission. This is $768.5004\; \pm \; 0.2030 \; mas$ (milliarcseconds) as for Gaia Data Release 2. This puts Proxima Centauri at a distance of $4.2441 \;\pm\; 0.0011\; ly$. Therefore the uncertainty is so small that the error spans $141.8 \; AUs$, which is more or less the size of the orbit of the Dwarf Planet Eris. The precission is staggering, but if we want to send one day a light-sail mission to this system and explore its planets we will need two orders of magnitude greater precission in the distance estimates (to at least start talking on how to aim at the $AU$ level).

NASA's New Horizons team has launched an interesting proposal to measure the parallax of Proxima Centauri and Wolf 359 by using the Long Range Reconnaissance Imager (LORRI) onboard of the New Horizons probe. Gaia was made to perform these astrometric measurements with high accuracy (less than a milliarcsecond) and New Horizons was not. But while Gaia has a $\sim 2 \; AU$ baseline for the parallax measurements, New Horizons is at $\sim 47 \; AU$ from Earth (as for the 6th February, 2020). The team wants to get this new measurement for Proxima's parallax on April 22, 2020, using simultaneus observations from Earth and the Kuiper Belt.

I guess that the real baseline for the measurement can't be the entire $47\; AU$ between the spacecraft and Earth but something less (since the probe is not moving in the plane perpendicular to the Proxima - Earth line), but still it will be a huge increase with respect to Gaia's baseline. Still LORRI is not as prepared as Gaia to measure small angles.

So, my question is, How these two things (larger baseline but lower ability to measure small angles for New Horizons) balance out? Will the distance to Proxima Centauri get more accurate with New Horizons parallax program, or is Gaia going to still hold the record for that measurement with its $2\; AU$ baseline? And if New Horizons gets a more accurate parallax measurement, how large we should expect the error in the distance to be compared to the current $141.8 \; AUs$ uncertainty measured by Gaia?
LORRI will be used in 4x4 mode, which yields 4-arcsec pixels.  The error in positions is unlikely to be better than 1%, or 40 mas, about 200x larger than Gaia's error.  NH has a baseline ~20x larger, but this means it still misses Gaia by ~10x.

The date was selected for New Moon to help Earth observers find the two targets.  There is no Earth analogue for LORRI, but it should be readily matched by amateur rigs.  The project will use professional ~0.5 - 1.0m telescopes to get Earth obs.  Simultaneous Earth obs are not strictly required, but the simultaneous times are offered to conduct the aesthetically purest demonstration of instantaneous parallaxes.