On a problem of Schinzel and Wójcik involving equalities between multiplicative orders

Given $a_1, \ldots, a_r \in\Q \{0, \pm 1\}$, the Schinzel–Wójcik problem is to determine whether there exist infinitely many primes $p$ for which the order modulo $p$ of each $a_1, \cdots, a_r$ coincides. We prove on the GRH that the primes with this property have a density and in the special case when each $a_i$ is a power of a fixed rational number, we show unconditionally that such a density is non zero. Finally, in the case when all the $a_i$'s are prime, we express the density it terms of an infinite product.