  We present a monomial ideal $\mathfrak{a} \subset S$ such that
$S/\mathfrak{a}$ is not Golod, even though the product on its Koszul homology
is trivial. This constitutes a counterexample to a well-known result by
Berglund and J\"ollenbeck (the error can be traced to a mistake in an earlier
article by J\"ollenbeck). On the positive side, we show that if $R$ is a
monomial ring such that the $r$-ary Massey product vanish for all $r \leq
\max(2, \mathrm{reg} R-2)$, then $R$ is Golod. In particular, if $R$ is the
Stanley-Reisner ring of a simplicial complex of dimension at most $3$, then $R$
is Golod if and only if the product on its Koszul homology is trivial.
Moreover, we show that if $\Delta$ is a triangulation of a $\Bbbk$-orientable
manifold whose Stanley-Reisner ring is Golod, then $\Delta$ is $2$-neighborly.
This extends a recent result of Iriye and Kishimoto.
