McKinnon's best-approximation curve conjecture

Let $X$ be a smooth projective variety defined over a number field $k$, let $P\in X(k)$, and let $A$ be an ample $\mathbb{Q}$-Cartier divisor on $X$. For an algebraic point $P$, write $\alpha(P,A)$ for the infimum of the approximation constants of sequences of $k$-rational points approaching $P$, and for a subvariety $i\colon Z\hookrightarrow X$ write $\alpha(P,A|_Z):=\alpha(P,i^*A)$. McKinnon's best-approximation curve conjecture. If $\alpha(P,A)<\infty$, then there is a curve $C$ of best $A$-approximation to $P$, namely

$$\alpha(P,A|_C)=\alpha(P,A).$$

This conjecture asserts that the optimal rational approximations to a rational point are controlled by a curve, and is the main approximation-theoretic statement studied in the paper.