Could very low-metallicity stars with rotation-dominated orbits have been shepherded by the bar?

The most metal-poor stars (e.g. [Fe/H] $\leq-2.5$) are the ancient fossils from the early assembly epoch of our Galaxy, very likely before the formation of the thick disc. Recent studies have shown that a non-negligible fraction of them have prograde planar orbits, which makes their origin a puzzle. It has been suggested that a later-formed rotating bar could have driven these old stars from the inner Galaxy outward, and transformed their orbits to be more rotation-dominated. However, it is not clear if this mechanism can explain these stars as observed in the solar neighborhood. In this paper, we explore the possibility of this scenario by tracing these stars backwards in an axisymmetric Milky Way potential with a bar perturber. We integrate their orbits backward for 6 Gyr under two bar models: one with a constant pattern speed and another one with a decelerating speed. Our experiments show that, under the constantly-rotating bar model, the stars of interest are little affected by the bar and cannot have been shepherded from a spheroidal inner Milky Way to their current orbits. In the extreme case of a rapidly decelerating bar, some of the very metal-poor stars on planar and prograde orbits can be brought from the inner Milky Way, but $\sim90\%$ of them were nevertheless already rotation-dominated ($J_{\phi}$ $\geq$ 1000 km s$^{-1}$ kpc) 6 Gyr ago. The chance of these stars having started with spheroid-like orbits with small rotation ($J_{\phi}$ $\lesssim$ 600 km s$^{-1}$ kpc) is very low ($