Functoriality of the motivic pro-semisimple completion

Let $(X,\tilde X)$ and $(X',\tilde X')$ be pairs as in the paper, with a morphism $(X',\tilde X')\to(X,\tilde X)$, and let $\hat f_{(\lambda)}^{\operatorname{mot}}:\widehat{\Pi'}^{\operatorname{mot}}\to\hat\Pi^{\operatorname{mot}}$ be the homomorphism induced by this morphism for a non-Archimedean place $\lambda$ not dividing $p$, well-defined up to $(\hat\Pi^{\operatorname{mot}})^\circ$-conjugation. Functoriality conjecture. The $(\hat\Pi^{\operatorname{mot}})^\circ$-conjugacy class of $\hat f_{(\lambda)}^{\operatorname{mot}}$ does not depend on $\lambda$. This asserts the expected compatibility of the motivic pro-semisimple completion with morphisms of smooth varieties; the supplied text gives no resolution status.