The flat Gauduchon connection conjecture

Let $(M^n,g)$ be a compact Hermitian manifold, and let $\nabla^s$ denote its $s$-Gauduchon connection. A connection is flat when its curvature tensor vanishes identically. Flat Gauduchon connection conjecture. If $s\neq 0,2$, then any compact Hermitian manifold $(M^n,g)$ which admits a flat $s$-Gauduchon connection must be Kähler, thus being a finite undercover of a flat complex torus. The cases $s=0$ and $s=2$ are understood through classifications of compact manifolds with flat Chern and Bismut connections, respectively; the proposed statement concerns the remaining Gauduchon connections.