  Plasma accelerators sustain extreme field gradients, and potentially enable
future compact linear colliders. Although tremendous progress has been achieved
in accelerating electron beams in a plasma accelerator, positron acceleration
with collider-relevant parameters is challenging. A recently proposed positron
acceleration scheme relying on the wake generated by an electron drive beam in
a plasma column has been shown to be able to accelerate positron witness beams
with low emittance and low energy spread. However, since this scheme relies on
cylindrical symmetry, it is possibly prone to transverse instabilities that
could lead, ultimately, to beam break-up. In this article, we show that the
witness beam itself is subject to various damping mechanisms and, therefore,
this positron acceleration scheme is inherently stable towards misalignment of
the drive and witness beams. This enables stable, high-quality plasma-based
positron acceleration.
