Automorphism conjecture for Hall algebra Drinfeld doubles

Let $\mathcal{C}$ be a hereditary finitary category. A derived functor is called friendly when it sends every object into two adjacent cohomological degrees. Automorphism conjecture. If every autoequivalence of $D^b(\mathcal{C})$ is friendly, then $\operatorname{Aut}(D^b(\mathcal{C}))$ acts on the reduced Drinfeld double $\widetilde{\mathbf{D}}\mathbf{H}_{\mathcal{C}}$ by algebra automorphisms. If the symmetrized Euler form $(\,\ )_a$ of $\mathcal{C}$ vanishes identically, then $\operatorname{Aut}(D^b(\mathcal{C})/T^2)$ acts by algebra automorphisms on the genuine Drinfeld double $\mathbf{D}\mathbf{H}_{\mathcal{C}}$. This is presented as a weaker consequence of the preceding two conjectures.