The reduction conjecture for Wintgen ideal submanifolds

Let $x:M^m\longrightarrow \mathbb{R}^{m+p}$ be a Wintgen ideal submanifold without umbilic points. Let $\mathbb{D}_2=\operatorname{Span}\{E_1,E_2\}$ be its canonical distribution, and suppose that it generates an integrable distribution $\mathbb{D}$ of dimension $k<m$. Reduction conjecture. Locally, $x$ is Möbius equivalent to a cone, a cylinder, or a rotational submanifold over a $k$-dimensional minimal Wintgen ideal submanifold in, respectively, ${\mathbb S}^{k+p}$, ${\mathbb R}^{k+p}$, or ${\mathbb H}^{k+p}$. This has been proved for $k=2$ and for $k=3,4,5$; the general case remains open.