Conjecture on extending the rational rotation-number height bound

Let $\lambda=a/b$ and $\delta=r/s$ be rational numbers with $0<\lambda,\delta<1$, and let $\gamma>1$. Theorem 2 concerns the rational rotation number $\rho_{\lambda,\delta}=p/q$ under the assumption $b>a^\gamma$. The conjecture. Theorem 2 should remain valid when $b>a^\gamma$, allowing a possibly larger upper bound for $q$ that depends only on $\gamma$, $a$, $b$, and $s$. This would extend the partial height estimate for rational rotation numbers beyond the range established with the golden-ratio exponent.