Three-weight code conjecture for Ding's odd-m difference sets

Let $m\geq 5$ be odd, let $D_f$ be defined from one of the eleven trinomials in Ding's difference-set conjecture, and let ${\mathcal C}_{D_f}$ be the associated binary linear code. Odd-m code conjecture. For every listed polynomial $f$, the code has parameters

$$[2^{m-1},\,m,\,2^{m-2}-2^{(m-3)/2}]$$

and weight enumerator

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

The conjecture predicts binary three-weight codes arising from the conjectured odd-dimensional cyclic difference sets.