Rigid-triple conjecture for exceptional finite groups

Let $\widehat{G}$ be of type $E_8$ or $G_2$, and let $C_1$ be the regular unipotent class in $\widehat{G}(\mathbb{F}_{\ell})$, $C_{\infty}$ the unipotent class of $v$, and $C_0$ the conjugacy class of the reduction of $\kappa$. A triple is strictly rigid when the equation

$$g_0g_1g_{\infty}=1,\qquad g_i\in C_i\quad\text{for }i=0,1,\infty$$

has a unique solution up to conjugacy and every such solution generates the whole group.

Rigid-triple conjecture. The triple $(C_0,C_1,C_{\infty})$ is strictly rigid in $\widehat{G}(\mathbb{F}_{\ell})$: the equation

$$g_0g_1g_{\infty}=1,\qquad g_i\in C_i\quad\text{for }i=0,1,\infty$$

has a unique solution up to conjugacy in $\widehat{G}(\mathbb{F}_{\ell})$, and any such solution generates $\widehat{G}(\mathbb{F}_{\ell})$.

Rigid triples are used in the inverse Galois problem to construct étale coverings and, via Hilbert irreducibility, Galois groups over $\mathbb{Q}$. The supplied text does not state whether this conjecture has been proved or refuted.