Ding's tri-weight code conjecture for trinomial value-sets

Let $m\ge 5$ be an odd integer. Let $f$ be any trinomial specified in the paper, let $D(f)^*$ be its punctured value-set, and let $\mathcal{C}_{D(f)^*}$ denote the associated binary trace code. Ding's tri-weight code conjecture. The code $\mathcal{C}_{D(f)^*}$ is a $[2^{m-1},m,2^{m-2}-2^{(m-3)/2}]$-code with weight enumerator polynomial

$$1+(2^{m-2}-2^{(m-3)/2})z^{2^{m-2}-2^{(m-3)/2}}+(2^{m-1}-1)z^{2^{m-2}}+(2^{m-2}+2^{(m-3)/2})z^{2^{m-2}+2^{(m-3)/2}}.$$

Furthermore, the dual code of $\mathcal{C}_{D(f)^*}$ is a $[2^{m-1},2^{m-1}-m,3]$-code. The abstract says that the paper gives a partial resolution of this conjecture, while the displayed formulation records the full claimed parameters and weight distribution.