Determination of a standard half-sided modular inclusion by its algebraic inclusion

Let $(\Omega,\mathcal N\subset\mathcal M)$ be a standard half-sided modular inclusion of factors. The normalization of $\Omega$ is allowed to be fixed separately. For another half-sided modular inclusion $(\tilde{\Omega},\tilde{\mathcal N}\subset\tilde{\mathcal M})$ with

$$\|\tilde{\Omega}\|=\|\Omega\|,$$

assume that there is a unitary operator $V$ such that

$$V\mathcal N V^*=\tilde{\mathcal N},\qquad V\mathcal M V^*=\tilde{\mathcal M}.$$

Determination conjecture. There exists a unitary operator $W$ such that

$$W\mathcal N W^*=\tilde{\mathcal N},\qquad W\mathcal M W^*=\tilde{\mathcal M},\qquad W\Omega=\tilde{\Omega}.$$

Thus, up to a possible normalization of the distinguished vector, the unitary equivalence class of a standard half-sided modular inclusion should be completely determined by the unitary equivalence class of the inclusion of factors. The conjecture is motivated by the preceding results for the associated Möbius covariant nets; the source does not establish it.