  In this paper, we introduce the notion of the strong Rees property (SRP) for
$\mathfrak{m}$-primary ideals of a Noetherian local ring and prove that any
power of the maximal ideal $\mathfrak{m}$ has its property if the associated
graded ring $G$ of $\mathfrak{m}$ satisfies $\text{depth} \ G \ge 2$. As its
application, we characterize two-dimensional excellent normal local domains so
that $\mathfrak{m}$ is a $p_g$-ideal. Finally we ask what
$\mathfrak{m}$-primary ideals have SRP and state a conjecture which
characterizes the case when $\mathfrak{m}^n$ are the only ideals which have
SRP.
