  Hydrodynamical simulations predict that a large amount of hydrogen (>0.1
solar masses) is removed from a hydrogen-rich companion star by the SN
explosion in the single-degenerate scenario of Type Ia supernovae (SNe Ia).
However, non-detection of hydrogen-rich material in the late-time spectra of
SNe Ia suggests that the hydrogen mass stripped from the progenitor system is
<0.001-0.058 solar masses. In this letter we include thermohaline mixing into
self-consistent binary evolution calculations for the helium-enriched
main-sequence (HEMS) donor channel of SNe Ia for the first time. We find that
the swept-up hydrogen masses expected in this channel are around 0.10-0.17
solar masses, which is higher than the observational limits, although the
companion star is strongly helium-enriched when the SN explodes. This presents
a serious challenge to the HEMS donor channel.
