High-weight generator polynomial conjecture for shortened second-order Reed–Muller codes

Let $m\ge 5$ be an integer, let $\alpha$ be a primitive element of ${\mathrm{GF}}(2^m)$, and let $g^*_{2,\alpha}(X)$ denote the generator polynomial of the shortened second-order Reed–Muller code ${\mathrm{SRM}}(2,m)$. High-weight generator polynomial conjecture. For every integer $m\ge 5$, there exists a primitive element $\alpha$ of ${\mathrm{GF}}(2^m)$ such that

$$\operatorname{wt}\left(g^*_{2,\alpha}(X)\right)>2^{m-2}.$$

The conjecture is motivated by numerical experiments on how the Hamming weight of $g^*_{2,\alpha}(X)$ depends on the choice of $\alpha$; the supplied text does not state a resolution.