Let $K$ be a number field, and let $C$ be an algebraic curve of genus $g \geq 2$ defined over $K$. We define $d_C$ to be the positive integer with the following property: there exists at most finitely many $\overline{\mathbb{Q}}$-points of $x \in C$ with the property that $[K(x) : K] \leq d_C$, and for any integer $d > d_C$, there exists infinitely many points $x \in C(\overline{\mathbb{Q}})$ such that $[K(x) : K] \leq d$. Note that this is well defined by Faltings' theorem. Here the notation $K(x)$ refers to adjoining all coordinates of $x$ to $K$. We say that $x \in C(\overline{\mathbb{Q}})$ is a *low degree point* of $C$ if $[K(x) : K] \leq d_C$. Denote by $S_C$ the set of low degree points.

We now consider the stable canonical height $h$ on $C(\overline{\mathbb{Q}})$ given by Arakelov theory and the canonical sheaf. By Bogomolov's conjecture (i.e., theorem of Shou-Wu Zhang and Emmanuel Ullmo), there exists a number $\varepsilon_C > 0$ such that the set $\{x \in C(\overline{\mathbb{Q}}) : h(x) < \varepsilon_C\}$ is finite, and for all $\delta > \varepsilon_C$, the set $\{x \in C(\overline{\mathbb{Q}}) : h(x) < \delta\}$ is Zariski-dense. Call the elements in $T_C = \{x \in C(\overline{\mathbb{Q}}) : h(x) < \varepsilon_C\}$ the set of *low height points*.

My question concerns the intersection of $T_C \cap S_C$, or points which are both low degree and low height. Clearly, this intersection can be empty: for example, when $C$ is a hyperelliptic curve we have $d_C = 1$, so $S_C = C(K)$, which can already be empty. Now suppose that $C(K) \ne \emptyset$. Do we still expect $S_C \cap T_C = \emptyset$ typically?