Functoriality conjecture for Drinfeld doubles in the Calabi–Yau case

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be finitary hereditary categories whose symmetrized Euler forms $(\,\ )_a$ vanish identically, so their Hall algebras have genuine Drinfeld doubles. Let $F:D^b(\mathcal{C}_1)/T^2\to D^b(\mathcal{C}_2)/T^2$ be exact and fully faithful. For each $M\in\mathcal{C}_1$, choose $N,N'\in\mathcal{C}_2$ such that

$$F(M)\simeq N\oplus N'[-1].$$

Calabi–Yau Drinfeld-double conjecture. The assignment

$$[M]^{\pm}\mapsto\nu^{-\langle N,N'\rangle_a}[N]^{\pm}\cdot[N']^{\mp}$$

extends to an embedding $F_*:\mathbf{D}\mathbf{H}_{\mathcal{C}_1}\hookrightarrow\mathbf{D}\mathbf{H}_{\mathcal{C}_2}$, and this embedding is an isomorphism if $F$ is a derived equivalence. This is the 2-periodic, genuine-Drinfeld-double analogue of the friendly-functor conjecture.