Target category for a full p-adic twistor correspondence

Identify and characterize the appropriate target category that should accompany the functorial construction of p-adic twistors, specifying the precise notion of "p-adic twistors" expected to form the counterpart of inscribed vector bundles with meromorphic integrable connection, in analogy with the p-adic Simpson correspondence.

Background

The paper constructs a fully faithful exact tensor functor from \mathbb{Q}_p-local systems to twistors on the relative thickened Fargues–Fontaine curve and discusses its relation to a conjecture of Fargues and Liu–Zhu about a functor to a category of p-adic twistors.

However, the authors note that, unlike the p-adic Simpson correspondence where the target category is clear (Higgs bundles over the base), in this p-adic twistor setting the correct counterpart category remains unclear. Pinpointing this category is necessary to complete the envisioned twistor correspondence and unify the constructions with broader expectations in p-adic Hodge theory.