The generalized Chen conjecture for proper biharmonic surfaces in four-dimensional space forms

Let $\varphi:M^2\to N^4(\epsilon)$ be a proper biharmonic immersion. Generalized Chen conjecture. Then $\epsilon>0$, so $N^4(\epsilon)$ is the four-dimensional sphere $\mathbb S^4(\epsilon)$, and the image $\varphi(M)$ lies minimally in the small hypersphere $\mathbb S^3(2\epsilon)$. This is identified in the source as the most important open problem in the topic. The constant-mean-curvature case follows from known results, while the non-CMC case is known for several subclasses, leaving the general assertion open.