A non-Golod ring with a trivial product on its Koszul homology

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.