The effective geometric Bogomolov conjecture for curves

Let $C$ be a smooth projective curve over $\overline{K}$ of genus at least $2$, let $J_C$ be its Jacobian, let $D$ be a degree-one divisor, let $j_D:C\to J_C$ be the associated embedding, and let $\lVert\cdot\rVert_{\mathrm{NT}}$ be the canonical Néron–Tate seminorm. Call $C$ isotrivial if it is obtained by base change from a curve over the constant field. Effective geometric Bogomolov conjecture for curves. If $C$ is non-isotrivial, then there exists $\epsilon>0$ such that

$$\{x\in C(\overline{K})\mid\lVert j_D(x)-P\rVert_{\mathrm{NT}}\leq\epsilon\}$$

is finite for every $P\in J_C(\overline{K})$. Moreover, if $C$ has a stable model over $\mathfrak{B}$, such an $\epsilon$ should be describable effectively in terms of geometric information from that stable model. This is explicitly stated as a stronger, effective form of the curve conjecture; its effectivity assertion remains open.