The strong Rickard conjecture for functor categories

Let $\mathcal{X}$ and $\mathcal{Y}$ be essentially small idempotent complete additive categories. Write $\operatorname{Mod}-\mathcal{X}$ for the category of modules over $\mathcal{X}$, $\operatorname{mod}_{\infty}-\mathcal{X}$ for its corresponding finitely presented functor category, $D$ and $D^-$ for the derived and bounded-above derived categories, $D^b$ for the bounded derived category, and $K^b(\mathcal{X})$ for the bounded homotopy category. Strong Rickard conjecture. The following conditions are equivalent:

$$\begin{array}{ll} \text{(0)} & D(\operatorname{Mod}-\mathcal{X})\text{ and }D(\operatorname{Mod}-\mathcal{Y})\text{ are triangle equivalent};\\ \text{(1)} & D^-(\operatorname{Mod}-\mathcal{X})\text{ and }D^-(\operatorname{Mod}-\mathcal{Y})\text{ are triangle equivalent};\\ \text{(2)} & D^-(\operatorname{mod}_{\infty}-\mathcal{X})\text{ and }D^-(\operatorname{mod}_{\infty}-\mathcal{Y})\text{ are triangle equivalent};\\ \text{(3)} & D^b(\operatorname{mod}_{\infty}-\mathcal{X})\text{ and }D^b(\operatorname{mod}_{\infty}-\mathcal{Y})\text{ are triangle equivalent};\\ \text{(4)} & K^b(\mathcal{X})\text{ and }K^b(\mathcal{Y})\text{ are triangle equivalent};\\ \text{(5)} & \text{There is a tilting subcategory }\mathcal{T}\text{ of }K^b(\mathcal{X})\text{ such that }\mathcal{T}\cong\mathcal{Y}\text{ as additive categories.} \end{array}$$

If $\mathcal{X}$ and $\mathcal{Y}$ are not assumed to be small, conditions (2), (3), (4), and (5) are still equivalent. Moreover, every triangle equivalence in (0)--(3) restricts to a triangle equivalence as in (4). This generalizes Rickard's Morita theory from rings to functor categories; partial results are known for more general small additive categories, but the full equivalence remains unresolved.