Equality of the two arithmetic Higgs functors

Let $F_1$, $F_2$, and $F_3$ be the functors defined above from enhanced log Higgs bundles to arithmetic Higgs bundles, with $F_3$ the direct construction and $F_2\circ F_1$ the construction through $G_K$-Higgs bundles. Functorial comparison conjecture. There is an isomorphism of functors

$$F_3\simeq F_2\circ F_1.$$

This asserts that the two constructions of an arithmetic Higgs bundle from an enhanced log Higgs bundle agree. The source states that the conjecture is true when ${\mathfrak X}={\operatorname{Spf}}({\mathcal O}_K)$; its status in general is not resolved there.