  We show that a compact complex manifold $X$ has no non-trivial nef
$(1,1)$-classes if there is a non-isomorphic bimeromorphic map $f\colon
X\dashrightarrow Y$ isomorphic in codimension $1$ to a compact K\"ahler
manifold $Y$ with $h^{1,1}=1$. In particular, there exist infinitely many
isomorphic classes of smooth compact Moishezon threefolds with no nef and big
$(1,1)$-classes. This contradicts a recent paper (Strongly Jordan property and
free actions of non-abelian free groups, Proc. Edinb. Math. Soc., (2022):
1--11).
