Extension of syntomic regulator formulae from integral models to curves

Let $F$ be the function field associated with the curve $C$, and let $\mathscr{C}$ denote the corresponding curve in place of the integral model $\mathcal{O}$. The preceding theorems give regulator formulae for $\mathcal{O}$ and $\mathscr{C}$. Extension conjecture. Theorems 1, 2, 3 and 4 should hold, with the same formulas, with $\mathcal{O}$ replaced by $F$ and $\mathscr{C}$ replaced by $C$. This would remove the integrality assumptions, which are needed only because the syntomic regulator is defined on the $K$-theory of an integral model.