Refined inverse Galois problem for abelian varieties

Fix a prime number $p$ and let $G\subset W_{2d}$ be a transitive subgroup containing the complex conjugation element. The weighted permutation representation associated to an abelian variety $A$ is the representation whose image is compared with $G$ below, and $w_A$ denotes the corresponding conjugating element. Refined inverse Galois conjecture. There exists an integer $r\geq 1$ and a simple abelian variety $A/\mathbf{F}_{p^r}$ of dimension $d$ such that $G$ is $w_A$-conjugate to the image of the weighted permutation representation associated to $A$, and $A$ may be taken to be ordinary. In particular, $G$ is isomorphic to the Galois group of some abelian variety $A$. This is presented as a slight refinement of a conjecture of Dupuy, Kedlaya, Roe, and Vincent concerning which groups arise as Galois groups of abelian varieties. The supplied context does not state whether the conjecture has been resolved.