The DDVV conjecture for submanifolds of space forms

Let $M^n$ be an immersed submanifold of the space form $N^{n+m}(c)$, with scalar curvature $\rho$, normal scalar curvature $\rho^{\perp}$, and mean curvature tensor $H=\frac{1}{n}\,\operatorname{trace}h$, where $h$ is the second fundamental form. DDVV conjecture. One has

$$\rho+\rho^{\perp}\leq |H|^2+c.$$

This is also called the normal scalar curvature conjecture. It is a central pointwise inequality in submanifold geometry and is equivalent to a matrix inequality for the coefficients of the second fundamental form. The supplied text does not state whether it had been resolved.