  We study the insulated conductivity problem with inclusions embedded in a
bounded domain in $\mathbb{R}^n$. The gradient of solutions may blow up as
$\varepsilon$, the distance between inclusions, approaches to $0$. An upper
bound for the blow up rate was proved to be of order $\varepsilon^{-1/2}$. The
upper bound was known to be sharp in dimension $n = 2$. However, whether this
upper bound is sharp in dimension $n \ge 3$ has remained open. In this paper,
we improve the upper bound in dimension $n \ge 3$ to be of order
$\varepsilon^{-1/2 + \beta}$, for some $\beta > 0$.
