Leydold's conjecture on maximal nodal domains of spherical harmonics

Let $\cal H_\ell$ be the eigenspace of spherical harmonics of degree $\ell$ on the sphere, and let $\mu(u)$ denote the number of nodal domains of a spherical harmonic $u$. Leydold's conjecture.

$$\max_{u\in \mathcal H_\ell}\mu(u)=\left\{\begin{array}{ll}\frac{1}{2}(\ell+1)^2 & \text{if }\ell\text{ is odd},\\[4pt]\frac{1}{2}\ell(\ell+2) & \text{if }\ell\text{ is even.}\end{array}\right.$$

The right-hand side is attained by decomposed spherical harmonics in spherical coordinates. The conjecture was proved for $\ell\leq 6$, but remains open in general; it would determine all possible maximal nodal-domain counts and has consequences for Courant-sharp eigenvalues and Pleijel-type bounds.