One-weight Griesmer code conjecture for repeated simplicial defining sets

Let $\mathbb{F}_{2^n}=\mathbb{F}_{2}(\eta)$, let $m\in\mathbb{N}$, and let $\emptyset\neq M\subsetneq[m]$. Set

$$D=\Delta_M+\eta\Delta_M+\cdots+\eta^{n-1}\Delta_M\subset\mathbb{F}_{2^n}^m.$$

Let $C_{D^{\ast}}$ be the associated punctured simplicial code, and let $A_i$ denote its weight distribution and $Z_i=|\{v\in\mathbb{F}_{2^n}^m:wt(c_{D^{\ast}}(v))=i\}|$. The repeated-set code conjecture. The code $C_{D^{\ast}}$ is a one-weight linear code over $\mathbb{F}_{2^n}$ of length $2^{n|M|}-1$, dimension $|M|$, and distance $(2^n-1)\times2^{n(|M|-1)}$; in particular it is minimal and is a Griesmer code. Moreover,

$$Z_0=2^{n(m-|M|)},\qquad Z_i=Z_0A_i.$$

The claim extrapolates computations for fields of orders $2$, $4$, $8$, and $16$; no general proof or resolution is supplied.