I'm reading the paper *Homogeneity
of the State Space of Factors of Type $\rm{III}_1$* by Connes and Størmer.
Homogeneity of the state space means that all normal states are approximately unitarily equivalent in the sense that for normal states $\psi,\phi$ on $M$ and $\epsilon>0$, there exists a unitary $u\in M$ such that
$$
\| u\psi u^* - \phi\|<\epsilon,
$$
where $u\psi u^* = \psi\circ\mathrm{Ad}_{u^*}$.

The proof of the implication "homogeneity implies type $\rm{III}_1$" confuses me. The proof seems to be reduced to Connes' *Une classification des facteurs de type $\rm{III}$* paper (which is in french).
Since type $\rm{III}_1$ is defined via the Connes invariant $S(M)$, which is the intersection of all modular spectra $\mathrm{Sp}\Delta_\psi$, they have to show that the modular spectrum is always $\mathbb R_+$. My question is:

*Can one prove directly that approximate unitary equivalence of two normal states $\phi,\psi$ on a factor $M$ implies that they have the same modular spectrum, i.e., $\mathrm{Sp}\Delta_\psi=\mathrm{Sp}\Delta_\phi$?*

If this were true, one could prove $\mathrm{Sp}\Delta_\psi=\mathbb R_+$ for all normal states $\psi$ using $M \cong M\otimes M_2(\mathbb C)$. Homogeneity would imply that $\psi\otimes \rho$ and $\psi$ have the same modular spectrum for all states $\rho$ on $M_2(\mathbb C)$ which can only hold if the spectrum is $\mathbb R_+$.

**Edit:** I think I might have an argument using ultrapower techniques but I'd still be very interested in a more direct argument.