A Nica-amenable submonoid of a group which fails to embed in an amenable group

We construct a weakly quasilattice-ordered submonoid of a group whose full semigroup ${\mathrm{C}}^{\ast }$-algebra is nuclear, while the monoid does not embed into any amenable group. Hence, Nica amenability, as well as the nuclearity of its semigroup C*-algebra, do not characterize embeddability into an amenable group for group-embeddable monoids.