Parameters of subfield complement codes over fields of characteristic 2

Let $D=\Delta_{M_1}+\eta\Delta_{M_2}+\cdots+\eta^{n-1}\Delta_{M_n}\subset\mathbb{F}_{2^n}^m$ be as above, with the hypotheses stated in the source, and let $C^{(2)}_{D^c}$ be the associated subfield complement code. The subfield complement parameters conjecture. The code $C^{(2)}_{D^c}$ is an $nm$-dimensional, two-weight linear code over $\mathbb{F}_2$ with distance

$$2^{nm-1}-2^{\sum_{j=1}^n|M_j|-1}.$$

Moreover, it is a Griesmer code and is minimal if $\sum_{j=1}^n|M_j|\leq nm-2$. The claim is based on computations for fields of orders $2$, $4$, $8$, and $16$; no general proof or resolution is supplied.