The geometric Sen morphism is the unique lift of the Kodaira--Spencer morphism

We show that the geometric Sen morphism of a de Rham torsor over a smooth rigid analytic variety over a $p$-adic field is the unique lift, along a natural map, of the Kodaira–Spencer morphism of the associated filtered torsor with integrable connection. This extends previous computations in the minuscule case, and implies that the geometric Sen morphism is the derivative of the lattice Hodge period map. The computation applies, in particular, to non-minuscule period domains generalizing local Shimura varieties, furnishing new examples of towers satisfying He’s stalkwise perfectoidness.