Conjecture on powers of characteristic-polynomial coefficients for twisted Carlitz modules

Let $q=2$. For integers $i,j,m,n$, let $H_{i,j,n}(m)$ be the corresponding coefficient polynomial and let $D(m,0),\ldots,D(m,i)$ be the previously defined polynomials, with $D(m,r)=0$ outside the specified index range. The ideal generated by them is

$$\langle D(m,0),\ldots,D(m,i)\rangle.$$

Powers-of-coefficients conjecture. For any $i,j,m,n$, there exists $\gamma\in\mathbb{N}$ such that

$$H_{i,j,n}^{\gamma}(m)\in\langle D(m,0),\ldots,D(m,i)\rangle.$$

Equivalently, a suitable power of each polynomial $H_{i,j,n}(m)$ lies in the ideal generated by $D(m,0),\ldots,D(m,i)$. The conjecture concerns the relation between the coefficient polynomials arising from the characteristic polynomial of the relevant matrix and the equations defining the loci of twists with prescribed analytic rank. Its resolution would clarify the algebraic structure of these rank loci in the case $q=2$; no resolution is supplied in the source.