The Coates–Sinnott conjecture in characteristic p

Let $K/k$ be the function-field extension, let $G=\operatorname{Gal}(K/k)$, and let $S$ and $\Sigma$ be the sets of places used above. Write $\Theta_{S,\Sigma}(u)$ for the associated equivariant $L$-function, and let $H_{\operatorname{et}}^2(O_{K,S},\mathbb Z_{\ell}(n))$ denote étale cohomology, for a prime $\ell\ne p$ and an integer $n\ge 2$.

Coates–Sinnott conjecture. With this notation, for every prime $\ell\ne p$ and every $n\in\mathbb Z_{\ge 2}$,

$$\Theta_{S,\Sigma}(q^{n-1})\in \operatorname{Ann}_{\mathbb Z_{\ell}[G]}\bigl(H_{\operatorname{et}}^2(O_{K,S},\mathbb Z_{\ell}(n))\bigr).$$

This is presented as the characteristic-$p$ analogue of the Coates–Sinnott conjecture. The supplied text gives no resolution status for this assertion, although the paper later proves a refined version involving Fitting ideals.