Uniqueness conjecture for canonical parallel translation

Let $X_K$ be a smooth connected scheme over $K$, let $x_0,x_1\in X(K)$, and let $F_{x_i}$ denote the corresponding fibre functors on the category under consideration. Write $K_{st}$ for the period extension and $Mor^{\otimes}(F_{x_0} \otimes K_{st};F_{x_1} \otimes K_{st})$ for the space of tensor-compatible morphisms between the resulting fibre functors. Theorem B asserts the existence of a canonical parallel translation

$$C_{x_0;x_1;X_K} \in Mor^{\otimes}(F_{x_0} \otimes K_{st};F_{x_1} \otimes K_{st})$$

satisfying the listed functoriality, composition, analyticity, extension, explicit logarithmic, and monodromy properties. Uniqueness conjecture. A collection

$$C_{x_0;x_1;X_K} \in Mor^{\otimes}(F_{x_0} \otimes K_{st};F_{x_1} \otimes K_{st})$$

satisfying those properties is unique. The conjecture says that the canonical parallel translation constructed in Theorem B is characterized uniquely by its stated properties; the source gives no resolution or further evidence for this uniqueness claim.