The integral approximation conjecture for weakly log Fano schemes

Let $(\mathfrak{X},\mathfrak{D})$ be a weakly log Fano scheme over the ring of integers $\mathfrak{o}$, with generic fiber $(X,D)$, and write $\mathfrak{U}=\mathfrak{X}\setminus\mathfrak{D}$. Assume that $(\mathfrak{X},\mathfrak{D})$ satisfies the integral Hilbert property. Let $x\in D(k)$ lie on a minimal stratum of $D=\mathfrak{D}_k$, let $v$ be an archimedean place, and let $L$ be an ample line bundle. A rational curve $\mathfrak{C}$ on $(\mathfrak{X},\mathfrak{D})$ is log rational if its complement by the boundary has at most one point at infinity, and toroidal if it has two points at infinity. Integral approximation conjecture. There exists a rational curve $\mathfrak{C}$ on $(\mathfrak{X},\mathfrak{D})$, either log rational or toroidal, such that

$$\alpha_v(x,\mathfrak{U};L)=\alpha_v(x,\mathfrak{C}_0;L).$$

The conjecture is an integral analogue of McKinnon's conjecture: under the weakly log Fano and integral Hilbert hypotheses, the best integral approximations to a boundary point should be governed by rational curves. The paper computes the relevant constants for the log rational and toroidal curves, but does not establish the asserted existence in general.