Extrinsic Berger conjecture for complex submanifolds of projective space

Let $M\hookrightarrow \mathbb{C} P^N$ be a complete, connected, and full complex submanifold, meaning that it is not contained in a proper hyperplane. Suppose its normal holonomy group is not the full unitary group of the normal space. Extrinsic Berger conjecture. Then $M$ has parallel second fundamental form. This conjecture is proposed as an extrinsic analogue of Berger's theorem; the paper proves the corresponding classification for complex submanifolds with parallel second fundamental form, but the stated implication remains unverified here.