For self-adjoint $A$ and $B$, when is $(A+iB)^*$ the closure of $A-iB$?

Question

Suppose that I have two self-adjoint operators $A$ and $B$ such that $\mathcal{D}(A)\cap\mathcal{D}(B)$ is dense. It is straightforward to show that $A\pm iB$ (with domains $\mathcal{D}(A)\cap\mathcal{D}(B)$) are closable. What are generic conditions so that $(A+iB)^*$ is the closure of $(A-iB)|_{\mathcal{D}(A)\cap\mathcal{D}(B)}$? $A$ or $B$ bounded suffices, but does this hold in general?