Conjecture on the primitivity of the over-exceptional lattice for K3 surfaces

Let $X$ be a K3 surface over an algebraically closed field. The over-exceptional lattice $E'(X)$ is the lattice generated by all irreducible curves orthogonal to every elliptic fibration inside the Picard lattice.

Primitivity conjecture for the over-exceptional lattice. If $X$ has infinite automorphism group, then the embedding

$$E'(X)\hookrightarrow\operatorname{Pic}(X)$$

is primitive.

This question is motivated by bounds on over-exceptional lattices and lattice-theoretic criteria for the weak Hilbert Property. The statement is refuted: the paper points to a counterexample when the automorphism-group hypothesis is absent.