$\DeclareMathOperator\ord{ord}$This is related to a question in the MO post, Does there exist a prime $p$ such that $\left|\frac{\mathrm{ord_{p}}(a)}{\mathrm{ord_{p}}(b)}-c\right|<\gamma$ for some small constant $\gamma$?

Let $p$ be a prime and $\ord_{p}(a)$ be the least positive integer $d$ such that $p\mid a^{d}-1$.

If $a$ and $b$ are two coprime natural numbers greater than 1, then does there exist a prime $p$ such that $ \frac{\ord_{p}(a)}{\ord_{p}(b)}>1$?

**Edit:**
Some progress on this problem has been made recently here.