= Solution
Let $v_1=(1,0)$ and $v_2=(1,m)$ generate the distinguished cone. If a complete fan had exactly one further ray with primitive generator $w=(a,b)$ and were smooth away from $x_\sigma$, the other two cones would be smooth. Completeness puts $w$ on the other side of the two boundary rays, so after choosing cyclic orientation $b=-1$. Smoothness would require
$$
\det(v_2,w)=-1-ma=1,
$$
hence $ma=-2$. This has no integer solution for $m\geq3$. Thus no proper toric variety can satisfy the three conditions from the PDF.
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