Is the flip-flop behaviour of accretion shock cones on to black holes an effect of coordinates?

We study numerically the relativistic BondiHoyle accretion of an ideal gas on to a Kerr fixed background spacetime on the equatorial plane with s-lab symmetry. We use both KerrSchild (KS) and BoyerLindquist (BL) coordinates. We particularly focus on the study of the flip-flop motion of the shock cone formed when the gas is injected at supersonic speed. The development of the flip-flop instability of the shock cone in the relativistic regime was reported recently for the first time. We reproduce the flip-flop behaviour found in the past when BL coordinates are used, and perform similar numerical experiments using horizon penetrating KS coordinates. We find that using KS coordinates the shock cone oscillates; however, such oscillations are not of the flip-flop type and their amplitude decreases with resolution.