Hsiang's uniqueness conjecture for Type E constant mean curvature hypersurfaces

Let $p,q\geq2$, and let a global solution curve $\gamma(t)=(x(t),y(t))$ of

$$(p+q-1)H=\frac{x'(t)y”(t)-x”(t)y'(t)}{(W(t))^3}+(q-1)\frac{y'(t)}{x(t)W(t)}-(p-1)\frac{x'(t)}{y(t)W(t)},$$

where $W(t)=\sqrt{(x'(t))^2+(y'(t))^2}$, generate an $O(p)\times O(q)$-invariant constant mean curvature $H$ hypersurface in $\mathbb{R}^{p+q}$. A Type E solution is one with exactly one cusp point at the origin. Hsiang's uniqueness conjecture. There is only one global solution curve of the equation above with one cusp point at the origin; equivalently, there is only one constant mean curvature $H$ hypersurface of $\mathbb{R}^{p+q}$, invariant under $O(p)\times O(q)$, with singularity at the origin. Hsiang's conjecture concerns the uniqueness of the singular Type E hypersurface among these invariant constant mean curvature hypersurfaces. The supplied source does not state whether the conjecture has been resolved.