Range of Self-Adjoint extensions

Question

$\newcommand{\Rg}{\operatorname{Rg}}$ Let $S$ be a densely defined symmetric operator on a complex Hilbert space $H$, with defect index $n_{+}=\dim(\Rg(S+i)^{\perp})=n_{-}=\dim(\Rg(S-i)^\perp)=m < \infty$. Let $T_1,T_2$ be self-adjoint extensions of $S$ and $\lambda$ $\in$ $\mathbb{C}$. Then if $\Rg(T_1-\lambda)$ is closed, so is $\Rg(T_2-\lambda)$.