  Let $M$ be a closed smooth Riemannian manifold $M$, and let $f:M\to M$ be a
diffeomorphism. Herein, we demonstrate that (i) if
  $f$ has the $C^1$ robustly inverse shadowing property on the chain recurrent
set $\mathcal{CR}(f)$, then $\mathcal{CR}(f)$ is hyperbolic and (ii) if $f$ has
the $C^1$ robustly inverse shadowing property on a nontrivial transitive set
$\Lambda\subset M$, then $\Lambda$ is hyperbolic for $f$. Especially, the item
(ii) is a proof of the conjecture of Lee and Lee \cite{LL}.
