The crossed-product factorization conjecture for Orlicz spaces

Let $\psi$ be an Orlicz function. The spaces $S^\psi$ and $S_\psi$ consist of the elements defined by the scaling conditions

$$S^\psi = \{a\in\widetilde{\mathcal{A}}: \theta_s(a\widetilde{\varphi}_{\psi^*}(h)^{1/2}\chi_{[0,\delta]}(h)) = e^{-s/2}a\widetilde{\varphi}_{\psi^*}(h)^{1/2}\chi_{[0,e^s\delta]}(h) \text{ for all } 0<\delta\}$$

and

$$S_\psi = \{a\in\widetilde{\mathcal{A}}: \theta_s(a{\varphi}_{\psi^*}(h)^{1/2}\chi_{[0,\delta]}(h)) = e^{-s/2}a{\varphi}_{\psi^*}(h)^{1/2}\chi_{[0,e^s\delta]}(h) \text{ for all } 0<\delta\}.$$

The crossed-product factorization conjecture. For any Orlicz function $\psi$,

$$L^\psi(\mathcal{M}) = \overline{\operatorname{span}}\{a^*b: a, b \in S^\psi\} \qquad\text{and}\qquad L_\psi(\mathcal{M}) = \overline{\operatorname{span}}\{a^*b: a, b \in S_\psi\}.$$

The author presents this relationship as currently unproved because of technical difficulties in establishing the asserted descriptions of the Orlicz spaces.