Conjecture that the crossed-product algebra is no longer type III

Let $\mathcal{M}$ be a local von Neumann factor, let $\sigma_s$ be its modular action, and let $\mathcal{R}(\mathcal{M},\sigma_s)$ be the algebra generated by $\mathcal{M}$ together with the modular evolution $\exp(isH_T)$. For a typical observable $\hat{B}$ of $\mathcal{R}(\mathcal{M},\sigma_s)$, let its spectral projectors be formed through the coupling of elements of $\mathcal{M}$ and $\exp(isH_T)$. Crossed-product type conjecture. Such spectral projectors are not related by partial isometries to projectors in, for example, $\mathcal{M}\otimes\mathbf{1}$ or to the identity projector $\mathbf{1}$, because their information contents differ. Hence $\mathcal{R}(\mathcal{M},\sigma_s)$ is no longer of type $III$. The paper explicitly says that this is not proved mathematically and argues for it on physical grounds. It is intended to explain how coupling a local algebra to Tomita evolution can change the type of the resulting algebra.