  We consider the insulated conductivity problem with two unit balls as
insulating inclusions, a distance of order $\varepsilon$ apart. The solution
$u$ represents the electric potential. In dimensions $n \ge 3$ it is an open
problem to find the optimal bound on the gradient of $u$, the electric field,
in the narrow region between the insulating bodies. Li-Yang recently proved a
bound of order $\varepsilon^{-(1-\gamma)/2}$ for some $\gamma>0$. In this paper
we use a direct maximum principle argument to sharpen the Li-Yang estimate for
$n \ge 4$. Our method gives effective lower bounds on the best constant
$\gamma$, which in particular approach $1$ as $n$ tends to infinity.
