Tate's symplecticity conjecture for the Brauer group of a surface

Let $X$ be a smooth, proper, geometrically connected surface over a finite field of characteristic $p$. Write

$$\operatorname{Br}(X):=\rH^2_{\et}(X;\G_m)$$

for its Brauer group, let $\operatorname{Br}(X)_{\nd}$ be the quotient by the subgroup of divisible elements, and let the Milne--Artin--Tate pairing be the non-degenerate skew-symmetric pairing on $\operatorname{Br}(X)_{\nd}$ constructed using Artin--Tate and Milne's methods. Tate's symplecticity conjecture. The Milne--Artin--Tate pairing is alternating. This conjecture predicts that the pairing is symplectic and concerns the refinement of the Artin--Tate construction from skew-symmetry to alternation; its status is not specified in the supplied text.