Artin's flat duality conjecture for surfaces

Let $\pi\colon X\rightarrow\operatorname{Spec} k$ be a smooth complete surface over a perfect field $k$ of characteristic $p\neq0$. Let $H_{\mathrm{fl}}^i(X,\mu_p)$ denote flat cohomology. Artin's flat duality conjecture. When $k$ is algebraically closed, there is a $4$-dimensional duality for the finite part of $H_{\mathrm{fl}}^i(X,\mu_p)$ and a $5$-dimensional duality for the vector-space part. The source says this rough formulation should be restated in derived-category terms and records that the precise statement is proved by Bragg and Olsson (2021).