Ding–Li conjecture on 2-designs from dual ternary codes

Let $m\ge 3$ be an odd integer, let $\alpha$ be a primitive element of $\operatorname{GF}(3^m)$, and let $\mathcal C_1$ and $\mathcal C_2$ be the ternary linear codes defined above. For $i\in\\{1,2\\}$, let $\mathcal C_i^{\perp}$ be the dual code, write $A_k(\mathcal C_i^{\perp})$ for the number of its codewords of weight $k$, and let $\mathcal P(\mathcal C_i^{\perp})$ and $\mathcal B_k(\mathcal C_i^{\perp})$ denote its point set and the supports of its weight-$k$ codewords. Ding–Li's dual 2-design conjecture. If $k$ is an integer satisfying $A_k(\mathcal C_i^{\perp})>1$, then

$$(\mathcal P(\mathcal C_i^{\perp}),\mathcal B_k(\mathcal C_i^{\perp}))$$

is a $2$-design. This extends the proposed design property to every nontrivial weight class of the dual codes; the supplied text gives no resolution status.