Haagerup and Størmer's conjecture for pointwise inner automorphisms

Let $M$ be a type $\rm III_1$ factor with separable predual. An automorphism $\theta\in\operatorname{Aut}(M)$ is pointwise inner if, for every positive linear functional $\varphi\in M_*^+$, there is a unitary $u\in\mathcal U(M)$ such that $\theta(\varphi)=u\varphi u^*$. A modular automorphism is an automorphism $\sigma_t^\varphi$ arising from Tomita–Takesaki theory for a faithful state $\varphi\in M_*^+$ and $t\in\mathbb R$. Haagerup and Størmer's conjecture. Every pointwise inner automorphism of $M$ is a composition of an inner automorphism and a modular automorphism. Inner and modular automorphisms are pointwise inner, so the conjecture asserts the converse for type $\rm III_1$ factors. The paper proves the assertion for factors with trivial bicentralizer and notes that it would hold in full generality if Connes' bicentralizer problem has an affirmative answer; the general conjecture is therefore left open.