  We prove that axially symmetric solutions to the $Q$-curvature type problem
$$ \alpha P_6 u + 120(1-\frac{e^{6u}}{\int_{\mathbb{S}^6} e^{6u}})=0 \ \ \ \ \
\mbox{on} \ \mathbb{S}^6 $$ must be constants, provided that $ \frac{1}{2}\leq
\alpha <1$. In view of the existence of non-constant solutions obtained by
Gui-Hu-Xie \cite{GHW2022} for $\frac{1}{7}<\alpha<\frac{1}{2}$, this result is
sharp. This result closes the gap of the related results in \cite{GHW2022},
which proved a similar uniqueness result for $\alpha \geq 0.6168$. The
improvement is based on two types of new estimates: one is a better estimate of
the semi-norm $\lfloor G\rfloor^2$, the other one is a family of refined
estimates on Gegenbauer coefficients, such as pointwise decaying and
cancellations properties.
