  A conjecture of Bombieri states that the coefficients of a normalized
univalent function $f$ should satisfy $$ \liminf_{f\to K} \frac{n-{\rm
Re\,}a_n}{m-{\rm Re\,}a_m} = \min_{t\in{\mathbb R}} \, \frac{n\sin t
-\sin(nt)}{m\sin t -\sin(mt)}, $$ when $f$ approaches the Koebe function
$K(z)=\frac{z}{(1-z)^2}$. Recently, Leung disproved this conjecture for $n=2$
and for all $m\geq3$ and, also, for $n=3$ and for all odd $m\geq5$.
Complementing his work we disprove it for all $m>n\geq2$ which are
simultaneously odd or even and, also, for the case when $m$ is odd, $n$ is even
and $n\leq \frac{m+1}{2}$. We mostly make use of trigonometry, but also employ
Dieudonn\'e's criterion for the univalence of polynomials.
