A K0-avoiding dimension group with an order-unit of index two

We prove that there exists a dimension group G whose positive cone is not isomorphic to the dimension monoid DimL of any lattice L. The dimension group G has an order-unit, and can be taken of any cardinality greater than or equal to 2. As to determining the positive cones of dimension groups in the range of the Dim functor, the 2 bound is optimal. This solves negatively the problem, raised by the author in 1998, whether any conical refinement monoid is isomorphic to the dimension monoid of some lattice. Since G has an order-unit of index 2, this also solves negatively a problem raised in 1994 by K.R. Goodearl about representability, with respect to K0, of dimension groups with order-unit of index 2 by unit-regular rings.