Derived invariance conjecture for reduced Drinfeld doubles

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be finitary hereditary connected $\mathbb{F}_q$-linear categories. Their reduced Drinfeld doubles are denoted by $\widetilde{\mathbf{D}}\mathbf{H}_{\mathcal{C}_1}$ and $\widetilde{\mathbf{D}}\mathbf{H}_{\mathcal{C}_2}$. Derived invariance conjecture. If $\mathcal{C}_1$ and $\mathcal{C}_2$ are derived equivalent, then

$$\widetilde{\mathbf{D}}\mathbf{H}_{\mathcal{C}_1}\cong\widetilde{\mathbf{D}}\mathbf{H}_{\mathcal{C}_2}$$

as algebras. This predicts that the reduced Drinfeld double is invariant under derived equivalence, with the stronger explicit functorial statement formulated in the following conjecture.