Dual minimum-distance conjecture for the code of the design ${\mathbb{D}}$

Let ${\mathbb{D}}$ be the design defined earlier in the paper, and let ${\mathsf{C}}_p({\mathbb{D}})^\perp$ denote the dual of its code over the field of characteristic $p$. Dual minimum-distance conjecture. The minimum distance of ${\mathsf{C}}_p({\mathbb{D}})^\perp$ equals

$$q+1.$$

The conjecture is motivated by experimental data. The surrounding theorem establishes only the lower bound $d^\perp\geq 3$ in general and identifies $d^\perp=p+1$ when $q=p$, so the asserted value remains unresolved in the supplied text for general $q$.