Morita invariance conjecture for the connected higher automorphism group

Let $\mathcal{C}$ be a graded $\Bbbk$-linear category such that

$$\operatorname{Hom}^i_{\mathcal{C}}(C,D)=0$$

for all $C,D\in\operatorname{Ob}(\mathcal{C})$ and all $i\geq 1$. Write $\operatorname{Aut}^{\infty}_{\circ}(\mathcal{C})$ for the connected component of the identity in the higher automorphism group of $\mathcal{C}$. Morita invariance conjecture. The group $\operatorname{Aut}^{\infty}_{\circ}(\mathcal{C})$ is a Morita invariant of $\mathcal{C}$. This would make precise the analogy with the connected component of the identity in the derived Picard group, assuming a suitable differential graded group-scheme interpretation of the derived Picard group. The source does not state a resolution of the conjecture.