J171 four-weight code conjecture

Let $m\geq 4$ be even, and define

$$f(x)=x+x^2+x^{2^m-2^{m/2}+1}\in\operatorname{GF}(2^m)[x],$$ $$D_f=\{f(x):x\in\operatorname{GF}(2^m)\}\setminus\{0\}.$$

J171 code conjecture. The binary code ${\mathcal C}_{D_f}$ has parameters

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

and has the weight distribution with nonzero weights and multiplicities

$$\begin{array}{c|c} \text{Weight} & \text{Multiplicity}\\ 2^{m-2}-2^{(m-2)/2} & 2^{(m-2)/2}\\ 2^{m-2}-2^{(m-4)/2} & 2^{m-1}-2^{m/2}\\ 2^{m-2} & 2^{m/2}+2^{(m-2)/2}-1\\ 2^{m-2}+2^{(m-4)/2} & 2^{m-1}-2^{m/2} \end{array}$$

The source describes this as a binary four-weight code conjecture and separately recalls the conjectured difference-set property of $D_f$.