Donagi–Pantev conjecture on derived categories of complementary elliptic fibrations

Let $X$ be a complex manifold elliptically fibered with at worst $I_{1}$ fibers over a normal analytic variety $B$ such that

$$H^{2}(B,\mathcal{O}^{\times})=\{1\}.$$

Let $\alpha,\beta\in\mathop{\mathcal{S}h}_{an}(X)$ be complementary elements. Write ${}_{\alpha}X_{\beta}$ and ${}_{\beta}X_{\alpha}$ for the corresponding twisted elliptic fibrations, and let ${\sf D}^{b}_{c}(-,w)$ denote the bounded derived category of constructible sheaves of weight $w$. Donagi–Pantev conjecture. There exists an equivalence

$${\sf D}^{b}_{c}({}_{\beta}X_{\alpha},-1)\cong {\sf D}^{b}_{c}({}_{\alpha}X_{\beta},1).$$

This extends the duality for complementary gerbes over elliptic fibrations; it was proven in many important special cases, while the general statement is presented here as a conjecture.