Cohomological rigidity conjecture for Airy local systems

Let $j:U\hookrightarrow X=\bP^1_k$ be an open immersion, let $u\in U$ be a geometric point, and let $\mathrm{Ai}_{\hG}$ be the Airy $\hG$-local system with corresponding representation $\rho:\pi_1(U,u)\to\hG(\bQ_l)$. Let $\widehat{\fg}^{\mathrm{der}}$ be the derived Lie algebra of $\widehat{\fg}$, let $\mathrm{Ai}_{\hG}^{\Ad}$ be the local system obtained through the adjoint representation, and let $n$ be the semisimple rank of $G$. Cohomological rigidity conjecture. The local system $\mathrm{Ai}_{\hG}$ is cohomologically rigid, meaning

$$\mathrm{H}^1\left(X,j_{!*}\mathrm{Ai}_{\hG}^{\Ad}\right)=0.$$

More specifically, the conjecture predicts

$$\mathrm{Sw}_{\infty}(\widehat{\fg}^{\mathrm{der}})=n(h+1),$$

and, if $\cI_{\infty}$ is the inertia group at $\infty$ in the Weil group $W_F$ of $F$, then

$$\left(\widehat{\fg}^{\mathrm{der}}\right)^{\rho(\cI_{\infty})}=0.$$

The paper presents these as predictions about the local structure of the Airy sheaves; it confirms the corresponding conjectures in the case $G=\GL_n$.