Arens–Michael envelope conjecture for invertible bimodules

Let $A$ be an algebra and let $M$ be an invertible $A$-bimodule, with inverse $M^{-1}$. Write $4\widehat{A}4$, $4\widehat{M}4$, and $4\widehat{M^{-1}}4$ for their Arens–Michael envelopes, and let $i_A$, $i_M$, and $i_{M^{-1}}$ denote the canonical maps. There should exist topological $A$-$\hat{\otimes}$-bimodule isomorphisms

$$\hat{i}_1:\widehat{M}\hat{\otimes}_{\widehat{A}}\widehat{M^{-1}}\longrightarrow\widehat{A},\qquad \hat{i}_2:\widehat{M^{-1}}\hat{\otimes}_{\widehat{A}}\widehat{M}\longrightarrow\widehat{A}$$

such that the following hold. Arens–Michael envelope conjecture. The bimodule $\widehat{M}$ is a topologically invertible $\widehat{A}$-$\hat{\otimes}$-bimodule with respect to $\hat{i}_1$ and $\hat{i}_2$, and the canonical maps make the multiplication diagrams commute: the map induced by $i_M\otimes i_{M^{-1}}$ intertwines $i_1$ with $\hat{i}_1$, while the map induced by $i_{M^{-1}}\otimes i_M$ intertwines $i_2$ with $\hat{i}_2$. This conjecture asks whether the Arens–Michael envelope of every invertible bimodule is topologically invertible, extending the preceding topological construction for automorphism bimodules. The source gives no resolution of the question.