Is there known a number field $K$ and a curve $F(x, y) \in K(t)[x, y]$ such that $F(x, y)$ does not have points over the field of rational functions $K(t)$ but for all but finitely many positive integer values of $t$ the respective specialization has a point over $K$?
(I expect the answer to be "such a $K$ is not known" and the problem of similar difficulty as that for $K = \mathbb{Q}$ but have not seen an expert comment on it so far.)
Thank you.