Non-1-convexity and non-n-convexity conjecture for ball bundles

Let $M$ be a compact Kähler manifold of dimension $k$, and let $E$ be a holomorphic $\mathbb{B}^n$-bundle over $M$ embedded in the associated $\mathbb{CP}^n$-bundle $\widehat{E}$. Write

$$E':= \widehat{E} \setminus \overline{E}.$$

A complex manifold is $q$-convex if it admits a smooth exhaustion function whose complex Hessian has at most $(q-1)$ non-positive eigenvalues outside a compact subset. The ball-bundle convexity conjecture. If $n<k$, then $E$ is not 1-convex and $E'$ is not $n$-convex. This extends the known partial results on convexity of ball bundles and predicts the obstruction when the fiber dimension is smaller than the base dimension.