The QALE conjecture for the hyperpolygon space $X_5(\theta)$

Let $X_5(\theta)$ be the hyperpolygon space associated with a generic parameter $\theta$. A smooth noncompact hyperkähler manifold is quasi-asymptotically locally Euclidean when its asymptotic geometry is modelled on $\mathbb{C}^{2k}/G$ for a finite subgroup $G$ of $\operatorname{Sp}(k)$ that need not act freely on the unit sphere. QALE conjecture. For generic $\theta$, the space $X_5(\theta)$ is quasi-asymptotically locally Euclidean. The case $X_4(\theta)$ is known to be asymptotically locally Euclidean, and the orbifold cone description of $X_5(0)$ provides evidence for the conjecture. The general asymptotic geometry and metric estimates remain to be established.