Arithmetic duality conjecture for finite extensions of L_p

Let $B$ be a finite extension of $L_p$. Write $H_{\operatorname{\text{\'et}}}^r(B;\mathbb{F}_{p^2}(m))$ for the étale cohomology groups with the indicated coefficients, and let $\cup$ denote the cup product. The proposed pairing takes values in $H_{\operatorname{\text{\'et}}}^3(B;\mathbb{F}_{p^2}(p+1))\cong\mathbb{Z}/p$.

Arithmetic duality conjecture. There is a perfect pairing

$$H_{\operatorname{\text{\'et}}}^r(B;\mathbb{F}_{p^2}(m))\otimes H_{\operatorname{\text{\'et}}}^{3-r}(B;\mathbb{F}_{p^2}(p+1-m))\overset{\cup}{\longrightarrow}H_{\operatorname{\text{\'et}}}^3(B;\mathbb{F}_{p^2}(p+1))\cong\mathbb{Z}/p$$

for each $r$ and $m$.

This extends the local Tate--Poitou duality pattern from $p$-adic fields to finite extensions of the periodic ring spectrum $L_p$. The source presents it as a conjectural arithmetic duality compatible with the motivic and étale cohomology structures.