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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.