Böhm–Wilking conjecture on invariant curvature cones

Let $n\geq 12$ and let $a\in[0,n/4]$. For a curvature operator $R\in S_B^2(\mathfrak{so}(n))$, write $\operatorname{Ric}$ for its Ricci tensor and $\operatorname{scal}$ for its scalar curvature. Define

$$\Omega(a)=\left\{R\in S_B^2(\mathfrak{so}(n))\ \middle|\ \left(\frac{n-2}{4}+a\right)\|R\|^2\leq\frac14|\operatorname{Ric}|^2,\ \operatorname{scal}>0\right\}.$$

Böhm–Wilking's conjecture. For every $a\in[0,n/4]$, the curvature cone $\Omega(a)$ is invariant under the Hamilton ODE and preserved by Ricci flow.

This conjecture proposes a family of Ricci-flow-invariant curvature conditions in dimensions $n\geq12$. The supplied text identifies it as unpublished and does not provide evidence of a resolution.