Vtyjkyuk

Given two complex numbers $u$ and $v$, numerically it seems that
$$1+vert uvert +vert vvertle vert 1+uvert +vert 1+vvert+vert u+vvert+vert 1+u+vverttag$*$$$
with equality holding for example when $(u=v=-1)$ Or $(u=exp(frac2ipi3),v=exp(frac4ipi3))$ but these are not the only cases.

This inequality does not follow in a clear way to me from Hlawka's inequality, but it looks some how similar to a difficult inequality
$$vert uvert +vert vvertle vert 1+uvert +vert 1+vvert+vert1+uvvert$$
which was proved in this paper.

I would appreciate any idea or help in proving $(*)$.

This follows from Hlawka. Denormalizing the inequality, you want to show
$$|x| + |y| + |z| leq |x+y| + |y+z| + |z+x| + |x+y+z|$$
Substitute $u = y+z, v = z+x, w = x+y$, you want to show
$$|v+w-u| + |w+u-v| + |u+v-w| leq 2(|u| + |v| + |w|) + |u+v+w|$$
But Hlawka tells you that
$$beginalign*
textRight hand side &ge (|u| + |v| + |w|) + (|u+v| + |v+w| + |w+u|) \
&= (|v+w| + |u|) + (|w+u| + |v|) + (|u+v| + |w|) \
&ge |v+w-u| + |w+u-v| + |u+v-w| \
&= textLeft hand side
endalign*$$