Ricci-flat Calabi's metric is not projectively induced

We show that the Ricci flat Calabi’s metrics on holomorphic line bundles over compact Kaehler-Einstein manifolds are not projectively induced. As a byproduct we solve a conjecture addressed in [A. Loi - F. Salis - F. Zuddas, Two conjectures on Ricci-flat Kaehler metrics, Math. Z. 290 (2018)] by proving that any multiple of the Eguchi-Hanson metric on the blow-up of C^2 at the origin is not projectively induced