Alternating-group Beauville structure conjecture for prescribed hyperbolic types

For positive integers $(r,s,t)$, call the type hyperbolic when

$$\frac{1}{r}+\frac{1}{s}+\frac{1}{t}<1.$$

Let $T(r,s,t)$ denote the corresponding hyperbolic triangle group. An unmixed Beauville structure is written $v=(a_1,c_1;a_2,c_2)$, with $(a_1,c_1)$ and $(a_2,c_2)$ having specified types.

Alternating-group Beauville structure conjecture. Let $(r,s,t)$ and $(r',s',t')$ be two hyperbolic types. Then almost all alternating groups ${\mathcal{A}}_n$ have an unmixed Beauville structure

$$v=(a_1,c_1;a_2,c_2)$$

where $(a_1,c_1)$ has type $(r,s,t)$ and $(a_2,c_2)$ has type $(r',s',t')$.

The claim is presented as suggestive, and the supplied status evidence resolves it in the context of the paper; the source also identifies it as a variation of Higman's conjecture, proved by B. Everitt in 2000.