  We identify minimal cases in which a power $m^i\not=0$ of the maximal ideal
of a local ring $R$ is not Golod, i.e.\ the quotient ring $R/m^i$ 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 $m^4=0\not= m^3$, the quotient
$R/m^3$ 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 $m^i\not= 0$ is Golod.
