Infinite families of harmonic self-maps of ellipsoids in all dimensions

We prove that for given $k\in \mathbb{N}$, $k\ge 3$, $d\in \mathbb{N}$ and each $a\in {\mathbb{R}}^{\ast }$ with \begin{align*} a^2<4d (d+k-2)(k-2)^{-2} \end{align*} the ellipsoid $E_a:=\{x\in {\mathbb{R}}^k\mid a^{-2}{x}_1^2+x_2^2+\dots +x_k^2=1\}$ admits infinitely many harmonic self-maps.