Functoriality conjecture for reduced Drinfeld doubles under friendly functors

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be finitary hereditary connected $\mathbb{F}_q$-linear categories. An exact functor $F:D^b(\mathcal{C}_1)\to D^b(\mathcal{C}_2)$ is friendly if there is an integer $n$ such that every $F(M)$ is concentrated in degrees $n$ and $n+1$. For an object $M$ of $\mathcal{C}_1$, write

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

with $N,N'\in\mathcal{C}_2$, set $\epsilon(n)=-(1)^n$, and define $\mathbf{d}(N,N',n)=n\langle N,N\rangle_a+(n+1)\langle N',N'\rangle_a-\langle N,N'\rangle_a$. Friendly-functor conjecture. If $F$ is exact, fully faithful, and friendly, then the assignments

$$\mathbf{k}_{\mathcal{F}}\mapsto\mathbf{k}_{N}^{\epsilon(n)}\mathbf{k}_{N'}^{-\epsilon(n)},$$ $$[M]^{\pm}\mapsto\nu^{\mathbf{d}(N,N',n)}[N]^{\pm\epsilon(n)}\mathbf{k}_{N}^{\pm n\epsilon(n)}\cdot[N']^{\mp\epsilon(n)}\mathbf{k}_{N'}^{\mp(n+1)\epsilon(n)}$$

extend to an embedding $F_*:\widetilde{\mathbf{D}}\mathbf{H}_{\mathcal{C}_1}\hookrightarrow\widetilde{\mathbf{D}}\mathbf{H}_{\mathcal{C}_2}$. This embedding is an isomorphism if $F$ is a derived equivalence. The conjecture gives an explicit derived-functorial realization of the preceding derived-invariance claim.