Local homogeneity conjecture for affine-type geometric structures

Let $X$ be a compact complex manifold with trivial canonical bundle, and let $\phi$ be a holomorphic geometric structure of affine type on $X$.

Local homogeneity conjecture. Any such structure $\phi$ is locally homogeneous. This implies that if $\phi$ is rigid, then the fundamental group of $X$ is infinite.

The conjecture generalizes the holomorphic Riemannian metric case and is motivated by results for holomorphic rigid geometric structures under algebraic-dimension hypotheses. It was proved in the supplied source's cited contexts when $X$ is Kähler or when its holomorphic tangent bundle is polystable with respect to a Gauduchon metric, but the general statement remains open.