Let $\varepsilon>0$ be sufficiently small. Denote by $\mathrm{Rm}$ and $\mathrm{R}$ the curvature operator and the scalar curvature.

Consider the following pinching condition $$\langle\mathrm{Rm}\,\phi,\phi\rangle\ge -\varepsilon\,\mathrm{R}\,|\phi|^2$$ for any tangent bivector $\phi$.

**Questions**

Is this piching preserved under Ricci flow? (This is true in 3-dimensional case, see Bing-Long Chen, Guoyi Xu, Zhuhong Zhang, Local pinching estimates in 3-dim Ricci flow.)

If yes, any references?

If no, is there any cone of curvature operators which is preserved under Ricci flow and just little wider than the cone $\mathrm{Rm}\ge 0$.