  We prove the unirationality of the Ueno-type manifold $X_{4,6}$. $X_{4,6}$ is
the minimal resolution of the quotient of the Cartesian product $E(6)^4$, where
$E(6)$ is the equianharmonic elliptic curve, by the diagonal action of a cyclic
group of order 6 (having a fixed point on each copy of $E(6)$). We collect also
other results, and discuss several related open questions.
