  In the early Universe, any particle carrying a conserved quantum number and
in chemical equilibrium with the thermal bath will unavoidably inherit a
particle-antiparticle asymmetry. A new particle of this type, if stable, would
represent a candidate for asymmetric dark matter (DM) with an asymmetry
directly related to the baryon asymmetry. We study this possibility for a
minimal DM sector constituted by just one (generic) $SU(2)_L$ multiplet $\chi$
carrying hypercharge, assuming that at temperatures above the electroweak phase
transition an effective operator enforces chemical equilibrium between $\chi$
and the Higgs boson. We argue that limits from DM direct detection searches
severely constrain this scenario, leaving as the only possibilities scalar or
fermion multiplets with hypercharge $y = 1$, preferentially quintuplets or
larger $SU(2)$ representations, and with a mass in the few TeV range.
