Weak Hilbert property conjecture for Néron models of elliptic curves

Let $K$ be a number field and let $Z\subset \mathbb{P}^1_K$ be an open subscheme, with $F=R(Z)$. Let $E_F$ be an elliptic curve over $F$ with semistable reduction along $Z$, or suppose that $\operatorname{rank}_{\mathbb{Q}}(E(F)\otimes_{\mathbb{Z}}\mathbb{Q})\geq 1$ and that $E_F$ has semistable reduction along $Z$. Let $f:E\to Z$ be the Néron model of $E_F$.

Néron-model WHP conjecture. Then $E$ has WHP over $K$.

The source calls this the most innocent-looking special case of the preceding open problem and states that it is currently unsolved.