  We prove that a word hyperbolic group whose Gromov boundary properly contains
a $2$-sphere cannot admit a projective Anosov representation into
$\mathsf{Sp}_{2m}(\mathbb{C})$, $m\in \mathbb{N}$. We also prove that a word
hyperbolic group which admits a projective Anosov representation into
$\mathsf{Sp}_{2m}(\mathbb{R})$ is virtually a free group or virtually a surface
group, a result established indepedently by Dey-Greenberg-Riestenberg.
