Ding–Li conjecture on Steiner systems 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, and let $\mathcal P(\mathcal C_i^{\perp})$ and $\mathcal B_4(\mathcal C_i^{\perp})$ denote its point set and the supports of its weight-$4$ codewords. Ding–Li's Steiner-system conjecture. Then

$$(\mathcal P(\mathcal C_i^{\perp}),\mathcal B_4(\mathcal C_i^{\perp}))$$

is a Steiner system $S(2,4,\frac{3^m-1}{2})$. The claim predicts Steiner systems obtained from the minimum-weight words of the dual codes; the supplied text gives no resolution status.