Classical-group Beauville structure conjecture with prescribed hyperbolic types

Let $(r_1,s_1,t_1)$ and $(r_2,s_2,t_2)$ be two hyperbolic types. Let $G$ be a finite simple classical group of Lie type, and let $(x_1,y_1;x_2,y_2)$ denote an unmixed Beauville structure, with the associated triples assigned their element orders.

Classical-group Beauville structure conjecture. If the Lie rank of $G$ is large enough, then $G$ admits an unmixed Beauville structure $(x_1,y_1;x_2,y_2)$ such that

$$(x_1,y_1,(x_1y_1)^{-1})$$

has type $(r_1,s_1,t_1)$ and

$$(x_2,y_2,(x_2y_2)^{-1})$$

has type $(r_2,s_2,t_2)$.

The statement proposes prescribed ramification types for sufficiently high-rank finite simple classical groups. The paper presents its results for low-rank families as initial evidence; the conjecture itself remains open.