Duality conjecture for entropic complexity of access structures

Let $P$ be a participant set, let $\mathcal A\subset 2^P$ be an access structure, and let $\mathcal A^\bot$ denote its dual access structure. Let $\sigma(\mathcal A)$ denote the entropic complexity of $\mathcal A$. Duality conjecture. For every access structure we have

$$\sigma(\mathcal A)=\sigma(\mathcal A^\bot).$$

The analogous equality is known for the polymatroid and linearly representable complexities, while the corresponding statement for entropic complexity is described as a long-standing open problem. The source explicitly says the conjecture is probably not true, so this conjecture is refuted.