Unitarity conjecture for simple current extensions of unitary vertex operator algebras

Let $W$ be a unitary vertex operator algebra, let $\pi\subset\mathrm{Rep}^\times(W)$ be a subgroup of its invertible unitary modules, and suppose that the simple current extension $\pi\ltimes W$ exists. Unitarity conjecture. For $W$ and $\pi$ as above, the simple current extension $\pi\ltimes W$ is unitary. This asserts that simple current extensions preserve unitarity, complementing the existence and uniqueness result for extensions whose relevant modules have integral $L_0$-eigenvalues. The supplied context does not establish whether this assertion is proved or remains open.