The Golod property of powers of the maximal ideal of a local ring

We identify minimal cases in which a power $m^i≠0$ of the maximal ideal of a local ring $R$ is not Golod, i.e.\ the quotient ring $R\mathrm{/}{m}^i$ is not Golod. Complementary to a 2014 result by Rossi and \c{S}ega, we prove that for a generic artinian Gorenstein local ring with $m^4=0≠m^3$, the quotient $R\mathrm{/}{m}^3$ is not Golod. This is provided that $m$ is minimally generated by at least $3$ elements. Indeed, we show that if $m$ is $2$-generated, then every power $m^i≠0$ is Golod.