The k-harmonic submanifold conjecture in Euclidean spaces

Let $x:M\rightarrow\mathbb{E}^n$ be an isometric immersion into a Euclidean space. For $k=1,2,\ldots$, call $x$ $k$-harmonic when

$$\bigtriangleup^k x=0,$$

equivalently $\bigtriangleup^{k-1}H=0$, where $H=-\frac{1}{m}\bigtriangleup x$ is the mean curvature vector and $\bigtriangleup$ is the Laplacian of $M$. The immersion is minimal when $H=0$.

The $k$-harmonic submanifold conjecture. The only $k$-harmonic submanifolds in Euclidean spaces are the minimal ones.

The claim extends Chen's biharmonic submanifold conjecture to all positive integers $k$. The paper proves the corresponding assertion for curves, but gives no resolution for general submanifolds.