The biharmonicity conjecture for non-CMC W-surfaces with flat normal bundle

Let $\varphi:M^2\to N^4(\epsilon)$ be a non-CMC W-surface with flat normal bundle. Assume that $3f^2+K-\epsilon\neq 0$ and $\nabla^\perp E_3\neq 0$ at any point. Biharmonicity conjecture. Then $M$ cannot be biharmonic. This conjecture is motivated by the preceding non-biharmonicity theorems for related classes of biconservative W-surfaces; its resolution is not established in the supplied text.