Haag duality conjecture for potentially non-unitary Wightman conformal field theories

Let $(\cF,\cD,U,\Omega)$ be a potentially non-unitary Wightman conformal field theory on $S^1$ with invariant nondegenerate Hermitian form. Let $\cQ(I)=\cP(I)”$ be the associated algebras, where the commutant is taken in $\operatorname{End}_*(\cD)$. Haag duality conjecture. The algebras $\cQ(I)$ are a Haag dual Möbius-covariant net of subalgebras of $\operatorname{End}_*(\cD)$; that is,

$$\cQ(I')=\cQ(I)'.$$

This asks whether the Haag duality familiar from unitary conformal field theory extends to potentially non-unitary Wightman theories; the supplied text does not state a resolution.