Meziane–Mortad conjecture on intertwining with a normal operator

Let $H$ be a Hilbert space, let $T$ be an operator on $H$, densely defined and closed if necessary, and let $B\in B(H)$ be normal. Meziane–Mortad's conjecture. If

$$BT\subseteq TB^*,$$

then

$$B^*T\subseteq TB.$$

This conjecture concerns a converse-type extension of the Fuglede–Putnam theorem to an operator $T$ that need not be bounded or self-adjoint. It was proposed by Meziane and Mortad and had resisted solutions for about three years at the time of this paper; its resolution is not established in the supplied text.