Existence and uniqueness of Frobenius-fixed paths in mixed motives

Let $Z$ be as in paragraph A16, let $X$ be the corresponding variety, and suppose that $x,y$ are $Z$-points of $X$. Let $s_\mathfrak{p}^*$ denote the specialization functor to the category of mixed motives over $k$, and let $\pi_1^\mathrm{un}(X,x,y)$ be the associated unipotent fundamental-group scheme object. The crystalline realization carries paths to de Rham paths, including the Frobenius-fixed de Rham path $p^\mathrm{Cr}$. Existence and uniqueness conjecture. There exists a unique morphism of affine scheme objects in $\operatorname{MM}(k)$

$$\mathbb{1} \to s_\mathfrak{p}^*\pi_1^\mathrm{un}(X,x,y)$$

whose crystalline realization maps via the Berthelot isomorphism to the Frobenius-fixed de Rham path $p^\mathrm{Cr}$. This is the expected extension beyond the mixed Tate setting of the corresponding proven statement in the semisimple mixed Tate category; it depends on the conjectural existence and standard properties of the relevant category of mixed motives and specialization functor.