The claimed detections of ⁶Li in the atmospheres of some metal-poor halo stars have lead to speculative additions to the standard model of Big Bang nucleosynthesis and the early Universe, as the inferred abundances cannot be explained by Galactic cosmic ray production. A prominent example of a so far un-disputed ⁶Li-detection is that of the metalpoor turn-off star HD84937, for which at least three different groups obtain very similar results. We revisit the lithium isotopic analysis for this star and two other halo stars, G64-12 and HD140283, using spectra of superiour quality and applying for the first time a combined 3D, NLTE synthesis to both the Li line and to Ca lines that are used to constrain the external line broadening. We demonstrate how differential NLTE effects between the Li and Ca line profiles can lead to spurious ⁶Li-detections in LTE, compatible with previously reported values of ⁶Li=⁷Li ≈ 5%. With our new analysis technique, none of the three analysed stars have a significant detection of ⁶Li in NLTE.We discuss which further progress is needed to firmly establish these results.