Global Chen conjecture for complete biharmonic submanifolds

Let $M$ be a complete biharmonic immersed submanifold of Euclidean space $\mathbb{E}^N$. Here, proper means that the preimage of every compact subset of $\mathbb{E}^N$ is compact in $M$; the global version concerns complete immersions, without requiring properness. A biharmonic submanifold has mean curvature vector field ${\bf H}$ satisfying

$$\Delta {\bf H}=0,$$

and it is minimal when ${\bf H}=0$.

Global Chen conjecture. Any complete biharmonic immersed submanifold in $\mathbb{E}^N$ is minimal.

This reformulation turns the question into one in global differential geometry. The source proves the stronger statement for complete proper biharmonic immersions, while the complete case without properness remains open.