Higher-singularity vanishing conjecture for twistor spaces

Let $X \subseteq \operatorname{iGr}(n,2n)$ be the big affine cell, or a ball or a polydisc inside it, and let $Z$ be the corresponding twistor space in the stated double fibration. For any coherent sheaf $\mathcal{F}$ on $Z$, consider its sheaf cohomology groups $H^k(Z,\mathcal{F})$. Higher-singularity vanishing conjecture. For all $k$ satisfying

$$k > s(n-s) - \frac{s(s-1)}{2},$$

we have

$$H^k(Z,\mathcal{F})=0.$$

This vanishing is proposed as the appropriate result for singularity size $|\Sigma|=s>1$ and would provide the cohomological input for extending the Penrose-transform construction to higher singularities; the source gives no resolution status.