Fixed-type Beauville structure conjecture for classical groups

Let $c\tau_1=(r_1,s_1,t_1)$ and $c\tau_2=(r_2,s_2,t_2)$ be two hyperbolic types. Let $G$ be a finite simple classical group of Lie type, with Lie rank sufficiently large. Fixed-type Beauville conjecture. The group $G$ admits an unmixed Beauville structure of type $(c\tau_1,c\tau_2)$. The analogous assertion for alternating groups is stated in the surrounding discussion as a theorem, while this classical-group version is presented as a conjecture.