Dervan–Sektnan's polystability conjecture for holomorphic fibrations

Let $\pi: (X,H)\to (B,L)$ be a holomorphic fibration, where $H$ and $L$ are the relevant polarisation classes. An optimal symplectic connection is a connection associated with the fibration whose curvature satisfies the optimal symplectic connection equation. Dervan–Sektnan's conjecture. The fibration $\pi: (X,H)\to (B,L)$ admits an optimal symplectic connection in $c_1(H)$ if and only if it is a polystable fibration. This is a Hitchin–Kobayashi correspondence-type conjecture relating the differential-geometric optimal symplectic connection equation to the conjectural algebro-geometric notion of stability for holomorphic fibrations.