Jones–Reutter's conjectural fiber sequence for monoidal 2-categories

Let $\mathfrak C$ be a monoidal $\mathbb K$-linear 2-category, subject to suitable assumptions. Write $\mathfrak C^\times$ for its subgroupoid of invertible objects, and let $\mathcal Aut_{\otimes}(\mathfrak C)$ and $\mathcal Out(\mathfrak C)$ denote the corresponding automorphism and outer automorphism groupoids. Jones–Reutter's conjecture. There is a homotopy fiber sequence

$$\mathfrak C^\times\to\mathcal Aut_{\otimes}(\mathfrak C)\to\mathcal Out(\mathfrak C).$$

This would categorify the established fiber sequence for monoidal categories and generalize the Rosenberg–Zelinsky exact sequence. The conjecture is presented as suggested by Corey Jones, but its suitable assumptions and general validity remain open.