Hyperplane Diameter Conjecture for bounded cells of arrangements

Let $\mathcal{A}$ be a simple arrangement of $n$ hyperplanes in dimension $d$, and let $H_\mathcal{A}(n,d)$ be the maximum average graph diameter of its bounded cells.

Hyperplane Diameter Conjecture.

$$H_\mathcal{A}(n,d) \leq d.$$

The conjecture concerns the average diameter of bounded cells in simple hyperplane arrangements. The supplied material gives no resolution status.