Boltyanski's Helly-dimension illumination bound

Let $\mathbf{K}$ be a convex body in $\mathbb{E}^d$, $d\geq 3$, and let $\operatorname{him}(\mathbf{K})$ be its Helly dimension, defined as the least $h$ for which the corresponding finite-family Helly property holds for translates of $\mathbf{K}$. Boltyanski's Helly-dimension conjecture. If $\operatorname{him}(\mathbf{K})=h>2$, then

$$I(\mathbf{K})\leq 2^d-2^{d-h}.$$

The case $h=2$ is stated as a theorem in the paper; the more general inequality is attributed to Boltyanski and remains open.