A $K\mathrm{\_}0$-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 Dim$L$ of any lattice $L$. The dimension group $G$ has an order-unit, and can be taken of any cardinality greater than or equal to $\mathrm{\aleph }\mathrm{\_}2$. As to determining the positive cones of dimension groups in the range of the Dim functor, the $\mathrm{\aleph }\mathrm{\_}2$ 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 two, this also solves negatively a problem raised in 1994 by K.R. Goodearl about representability, with respect to $K\mathrm{\_}0$, of dimension groups with order-unit of index 2 by unit-regular rings.