  We consider an insulated conductivity model with two neighboring inclusions
of $m$-convex shapes in $\mathbb{R}^{d}$ when $m\geq2$ and $d\geq3$. We
establish the pointwise gradient estimates for the insulated conductivity
problem and capture the gradient blow-up rate of order
$\varepsilon^{-1/m+\beta}$ with
$\beta=[-(d+m-3)+\sqrt{(d+m-3)^{2}+4(d-2)}]/(2m)\in(0,1/m)$, as the distance
$\varepsilon$ between these two insulators tends to zero. In particular, the
optimality of the blow-up rate is also demonstrated for a class of axisymmetric
$m$-convex inclusions.
