Conjecture on the weight distribution of a binary linear code from a two-to-one polynomial

Let $n=2m+1$ and let

$$f(x)=x^{3\cdot 2^{m+1}}+x^{2^{m+2}+1}+x^{2^{m+1}+1}+x.$$

Suppose that $f$ is considered over ${\mathbb F}_{2^n}$, and let $\mathcal{C}_{D(f)}$ denote the associated linear code. When $m\geq 4$, the conjectured parameters and possible codeword weights are

The weight-distribution conjecture. The polynomial $f(x)$ is two-to-one over ${\mathbb F}_{2^n}$, the code $\mathcal{C}_{D(f)}$ has parameters

$$\left[2^{n-1}-1,n,2^{n-1}-2^{\frac{n-1}{2}}\right],$$

and its codeword weights satisfy

$$\mathrm{wt}(\mathbf{c}_b)\in\left\{2^{n-2},0,2^{n-2}-2^{\frac{n-3}{2}},2^{n-2}+2^{\frac{n-3}{2}},2^{n-2}-2^{\frac{n-1}{2}},2^{n-2}+2^{\frac{n-1}{2}}\right\}.$$

The remaining problem is to determine the weight distribution of $\mathcal{C}_{D(f)}$.