Hasse principle for Kummer varieties in the case of generic 2-torsion

Conditional on finiteness of relevant Shafarevich--Tate groups, Harpaz and Skorobogatov established the Hasse principle for Kummer varieties associated to a 2-covering of a principally polarised abelian variety A, under certain large image assumptions on the Galois action on A[2]. However, their method stops short of treating the case where the image is the full symplectic group, due to the possible failure of the Shafarevich--Tate group to have square order in this setting. I will explain work in progress which overcomes this obstruction by combining second descent ideas of Harpaz with new results on the 2-parity conjecture.