 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 requirehence $ma=-2$. This has no integer solution for $m\ge 3$. Thus no proper toric variety can satisfy the three conditions from the PDF.