Fuglede–Putnam conjecture for unbounded operators

Let $H$ be a complex Hilbert space. Let $A$ be an operator on $H$, densely defined and closed if necessary, and let $B\in B(H)$ be normal. Fuglede–Putnam conjecture. The inclusion

$$BA\subset AB^*$$

should imply

$$B^*A\subset AB.$$

This extends the Fuglede–Putnam theorem from bounded $A$ to the stated unbounded-operator setting. The claim is known when $A\in B(H)$, but the source states that it is not covered by known unbounded generalizations.