Potential good reduction conjecture for basic admissible pairs

Let $L/\mathbb{Q}_p$ be a $p$-adic field and let $X/L$ be a smooth rigid analytic variety. Let $\mathrm{AdmPair}^{\mathrm{basic},\mathrm{pgr}}(X^\diamond)$ denote basic admissible pairs with potentially good reduction, and let $\mathrm{VHS}(X^\diamond)$ denote variations of $p$-adic Hodge structure. Potential good reduction conjecture. The functor

$$\mathrm{AdmPair}^{\mathrm{basic},\mathrm{pgr}}(X^\diamond)\longrightarrow\mathrm{VHS}(X^\diamond)$$

is essentially surjective and therefore defines an equivalence

$$\mathrm{AdmPair}^{\mathrm{basic},\mathrm{pgr}}(X^\diamond)\xrightarrow{\sim}\mathrm{VHS}(X^\diamond).$$

For $X=\operatorname{Spa}L$, the conjecture holds by the usual potentially crystalline equivalence; the general variation statement remains open in the supplied text.