Convexity conjecture for the core

The notation $obreak\mathrm{Core}(i)$ denotes the set of derivative directions associated with index $i$, and the empty set is regarded as convex.

Convexity conjecture.

$$\mathrm{Core}(i)\text{ is convex}.$$

The preceding theorem shows that, for $e\in\mathrm{Core}(i)$, the minimal affine space containing $e$ and the optimal set remains inside $\mathrm{Core}(i)$ after intersection with the positive-positive region. The conjecture would establish the stronger global convexity property, but no proof or counterexample is given here.