Non-Zariski-dense best-approximation 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 line bundle on $X$. Let $\alpha(P,A)$ denote the infimum of approximation constants of sequences of $k$-rational points approaching $P$, and let $\alpha_{ess}(P,A)$ denote the corresponding infimum over Zariski-dense sequences. For a subvariety $i\colon V\hookrightarrow X$, write $\alpha(P,A|_V):=\alpha(P,i^*A)$. Non-Zariski-dense best-approximation conjecture. If $\alpha(P,A)<\infty$, then there is a sequence of best $A$-approximations to $P$ that is not Zariski dense. Equivalently, there is a proper subvariety $V$ such that

$$\alpha(P,A|_V)\leq\alpha_{ess}(P,A).$$

This is a weaker higher-dimensional analogue of the curve conjecture: unlike for surfaces, the argument only guarantees a proper subvariety rather than a curve.