The centre conjecture for Mackey functors of a lextensive finite coproduct completion

Let $\mathcal{C}$ be a small category such that its finite coproduct completion $\operatorname{fam}\mathcal{C}$ is lextensive. Let $\operatorname{Aut}\mathcal{C}$ be the category of automorphisms in $\mathcal{C}$, whose objects are pairs $(X,u)$ with $u:X\to X$ invertible in $\mathcal{C}$. Write $\operatorname{Mky}_k$ for the category of $k$-linear Mackey functors and $\mathcal{Z}$ for the centre of a category. The centre conjecture. There is an equivalence

$$\mathcal{Z}\operatorname{Mky}_k(\operatorname{fam}\mathcal{C})\simeq\operatorname{Mky}_k(\operatorname{fam}\operatorname{Aut}\mathcal{C}).$$

This conjecture generalizes Tambara's result for the category of finite $G$-sets, where the centre of the category of $k$-linear Mackey functors is again a category of Mackey functors associated with a suitable category. The source provides no resolution of the conjecture.