Complement-code parameters conjecture over fields of characteristic 2

Let $\mathbb{F}_{2^n}=\mathbb{F}_{2}(\eta)$ with $\eta\in\mathbb{F}_{2^n}$, and let

$$D=\Delta_{M_1}+\eta\Delta_{M_2}+\cdots+\eta^{n-1}\Delta_{M_n}\subset\mathbb{F}_{2^n}^m,$$

where $m\in\mathbb{N}$, $\emptyset\neq M_i\subsetneq[m]$ for $1\leq i\leq n$, and $\bigcup_{i=1}^nM_i\subsetneq[m]$. Let $C_{D^c}$ be the associated complement code. The complement-code parameters conjecture. If $C_{D^{\ast}}$ is a $t$-weight linear code over $\mathbb{F}_{2^n}$, then $C_{D^c}$ is an $m$-dimensional $(t+1)$-weight linear code over $\mathbb{F}_{2^n}$ with distance

$$(2^n-1)\times\left(2^{n(m-1)}-2^{\sum_{j=1}^n|M_j|-n}\right).$$

It is also a Griesmer code, and it is minimal whenever $\sum_{j=1}^n|M_j|\leq nm-(n+1)$. The statement is based on computations for fields of orders $2$, $4$, $8$, and $16$, with no proof supplied.