Haagerup and Størmer's conjecture for pointwise inner automorphisms

In 1988, Haagerup and Størmer conjectured that any pointwise inner automorphism of a type $\mathrm{I}\mathrm{I}{\mathrm{I}}_1$ factor is a composition of an inner and a modular automorphism. We study this conjecture and prove that any type $\mathrm{I}\mathrm{I}{\mathrm{I}}_1$ factor with trivial bicentralizer indeed satisfies this condition. In particular, this shows that Haagerup and Størmer’s conjecture holds in full generality if Connes’ bicentralizer problem has an affirmative answer. Our proof is based on Popa’s intertwining theory and Marrakchi’s recent work on relative bicentralizers.