Euclidean depth of the real quadratic field of discriminant 30

Prove or refute Cooke's conjecture that the Euclidean depth of the ring of integers of $\mathbb{Q}(\sqrt{30})$ equals $2$.

Background

The paper gives an upper bound of three for the Euclidean depth of $D(30)$ and reports that Cooke conjectured the sharper value two. The table and surrounding discussion treat the depth as unresolved at that point.