The Carlitz zeta value Fitting-ideal conjecture for negative twists

For an integer $n$ written as $n=\delta+h(q-1)$ by Euclidean division by $q-1$, define

$$Z(x,-n)=\sum_{d\geq 0}x^dS_d(n),\qquad S_d(n)=\sum_{\substack{a\ \text{monic}\\ \deg(a)=d}}a(t)^n,$$

and let $P_n(x)$ be

$$P_n(x):=\operatorname{det}_{\mathbb{F}[t]}\left(\operatorname{id}-(\operatorname{id}-\gamma)x\bigg|\bigoplus_{k=0}^{h-1}\mathbb{F}[t]\cdot\theta^k\nu_n\right)\in\mathbb{F}[t][x].$$

Let $\zeta_C^*(-n)$ be the first nonzero coefficient in the expansion of $Z(x,-n)$ at $x=1$. Carlitz zeta value Fitting-ideal conjecture. We have $P_n(x)=Z(x,-n)$. In particular, $\zeta_C^*(-n)$ generates the Fitting ideal of $\operatorname{Cl}(\underline{A}(-n))$. The conjecture identifies the determinant polynomial arising from the class-module computation with the Carlitz zeta polynomial, and would therefore express the Fitting ideal of the class module in terms of a special Carlitz zeta value; it was motivated by numerical computations.