The largest xeric open set conjecture

Let $k$ be a number field and let $X$ be a smooth projective geometrically irreducible variety over $k$. Let $Z_{rc}(X)\subseteq X$ be the Zariski closure of the union of all possibly singular rational curves in $X_{k^{\mathrm{alg}}}$. Largest xeric open set conjecture. The open subset $X\setminus Z_{rc}(X)$ is xeric and hence is the largest xeric open set in $X$. This conjecture is suggested by Manin's rational curve conjecture and gives a geometric description of the maximal open locus where rational points are sparse. Its status is open.