Rational points conjecture for generic marked hypersurfaces

Let $k$ be a field, let $m\geq0$ be an integer, and let $X_m/K_m$ be the generic $m$-marked degree-$d$ hypersurface in $\mathbb{P}^{n+1}_k$. The points $\delta_1,\ldots,\delta_m$ denote its $m$ tautological sections.

Rational points conjecture. If $d\geq4$, then

$$X_m(K_m)=\{\delta_1,\ldots,\delta_m\}.$$

This predicts that a generic degree-$d$ hypersurface with $m$ marked points has no $K_m$-rational points beyond the tautological marked points. The surrounding discussion presents it as the optimal expectation for the rational-point theorem in the paper; no resolution is supplied here.