A problem in comparative order theory

Write $$\mathrm {ord}_p(\cdot )$$ for the multiplicative order in $${\mathbb {F}}_p^{\times }$$ . Recently, Matthew Just and second author investigated problem of classifying pairs $$\alpha , \beta \in {\mathbb {Q}}^{\times }\setminus \{\pm 1\}$$ which {ord}_p(\alpha ) > \mathrm {ord}_p(\beta holds infinitely many primes p. They called such order-dominant. We describe an easily-checkable sufficient condition ,\beta $$ to be Via large sieve, we show that almost all integer satisfy our condition, with a power savings on size exceptional set.