Generalized cohomology vanishing conjecture for parabolic connections

For $x\in\mathbb{P}^1$, let $\xi_x$ be the bundle on $\overline{\mathcal{M}}$ whose fiber at $(L,\nabla,\varphi;\epsilon\in E)$ is $L_x$. Let $\mathcal{M}$ denote the moduli space appearing in the construction. Generalized cohomology vanishing conjecture. Suppose $n\geq 4$, $x_1,\dots,x_{2(n-3)}\in\mathbb{P}^1$, and $x_i\neq x_j$ for $i\neq j$. Then

$$H^i(\mathcal{M},\xi_{x_1}\otimes\cdots\otimes\xi_{x_{2(n-3)}})=0$$

for any $i\geq 0$. The $n=4$ case is known from the cited result, and the $n=5$ case is the theorem proved in the paper; the general case is predicted to hold.