Ding–Li conjecture on 2-designs from ternary linear 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 by

\mathcal{C}_1=\left\\{ \left( \operatorname{Tr}_{3^m/3}\left( a \alpha^{4i}+b \alpha^{2i} \right) \right)_{i=0}^{\frac{3^m-1}{2}-1}:a,b\in \operatorname{GF}(3^m)\right\\},

and

\mathcal{C}_2=\left\\{ \left( \operatorname{Tr}_{3^m/3}\left( a \alpha^{\left(3^{\frac{m-3}{2}}+1\right)i}+b \alpha^{\left(3^{\frac{m-1}{2}}+1\right)i} \right) \right)_{i=0}^{\frac{3^m-1}{2}-1}:a,b\in \operatorname{GF}(3^m)\right\\},

where $\operatorname{Tr}_{3^m/3}$ is the trace from $\operatorname{GF}(3^m)$ to $\operatorname{GF}(3)$. For $i\in\\{1,2\\}$, write $A_k(\mathcal C_i)$ for the number of codewords of weight $k$, and let $\mathcal P(\mathcal C_i)$ and $\mathcal B_k(\mathcal C_i)$ denote the point set and the supports of the weight-$k$ codewords, respectively. Ding–Li's 2-design conjecture. If $k$ is an integer satisfying $A_k(\mathcal C_i)>1$, then

$$(\mathcal P(\mathcal C_i),\mathcal B_k(\mathcal C_i))$$

is a $2$-design. This is one of the conjectured infinite families of $2$-designs arising from linear projective ternary codes; the supplied text gives no resolution status.