  In this paper we show that for an $\text{Sp}(k+1)$ invariant metric $\hat{g}$
on $\mathbb{S}^{4k+3}$ $(k\geq 1)$ close to the round metric, the conformally
compact Einstein (CCE) manifold $(M, g)$ with $(\mathbb{S}^{4k+3}, [\hat{g}])$
as its conformal infinity is unique up to isometries. Moreover, by the result
in [LiQingShi], $g$ is the Graham-Lee metric on the unit ball $B_1\subset
\mathbb{R}^{4k+4}$. We also give an a priori estimate on the Einstein metric
$g$. Based on the estimate and Graham-Lee and Lee's seminal perturbation
result, we use the continuity method directly to obtain an existence result of
the non-positively curved CCE metric with prescribed conformal infinity
$(\mathbb{S}^{4k+3}, [\hat{g}])$ when the metric $\hat{g}$ is
$\text{Sp}(k+1)$-invariant. We also generalize the results to the case of
conformal infinity $(\mathbb{S}^{15},[\hat{g}])$ with $\hat{g}$ a
Spin$(9)$-invariant metric in the appendix.
