The pinching conjecture for totally geodesic and Veronese submanifolds

Let $M$ be the submanifold under consideration, let $\sigma$ denote its second fundamental form, and let $n$ be its dimension. Pinching conjecture. There is a constant $\varepsilon(n)$, depending only on $n$, such that if

$$\lVert\sigma\rVert^2\leq \frac{2}{3}n+\varepsilon(n),$$

then $M$ has to be totally geodesic or a Veronese surface in $S^4$. This is a proposed strengthening of pinching results discussed after the DDVV inequalities. The supplied text does not define the ambient setting beyond the stated conclusion or give evidence of resolution.