Parameter-independence conjecture for umbilic indices on superellipsoids

Let $a,b,c>0$ and let $k\in\mathds{Z}_{>1}$. Consider the surface

$$ax^{2k}+by^{2k}+cz^{2k}=1.$$

An umbilic point is a point where the principal curvatures coincide, and its index is the index of the principal direction field around that point. Parameter-independence conjecture. The indices of the umbilic points on this surface are independent of $a$, $b$, $c$, and $k$. The conjecture is motivated by numerical experiments on the displayed family, but no proof or resolution is supplied in the source.