Deligne's conjecture on Frobenius invariants of lisse sheaves with fixed local monodromy

Let $C$ be a smooth curve over a finite field $\mathbb F_q$, let $\bar C$ be its smooth compactification, and let $M$ be the set of irreducible lisse $\bar{\mathbb Q}_\ell$-sheaves of rank $r$ on $C\otimes_{\mathbb F_q}\mathbb F$ with prescribed local data. Define

$$2d=\dim H^1(\bar C\otimes_{\mathbb F_q}\mathbb F,j_*\operatorname{End}(V)).$$

Let $N(n)$ denote the cardinality of the $F^n$-invariants of $M$. Deligne's conjecture. (i) There are finitely many Weil numbers $a_i,b_j$ of weights between $0$ and $2d$ such that

$$N(n)=\sum_i a_i^n-\sum_j b_j^n.$$

(ii) If $M\neq\emptyset$, precisely one of the numbers $a_i,b_j$ has weight $2d$; it is one of the $a_i$ and equals $q^d$. The conjecture predicts a precise Weil-number description of the point counts of the moduli problem with fixed local monodromy. The supplied text gives the geometric setup and the asserted formula but no resolution status.