Question: suppose that $H_{n_1}, H_{n_2}, H_{n_3} \in L^{2}(\mathbb{S}^2)$ are Spherical Harmonics of degrees $n_j$ $(j = 1, 2, 3)$ with $n_1 > n_2 + n_ 3$. Then, it is true that $$ \int_{\mathbb{S}^2} H_{n_1} H_{n_2} H_{n_3} dS = 0 \; ? $$

Obs.: My failed attempt to solve this question was to show that $H_{n_2} H_{n_3}$ is a sum of Spherical Harmonics $H_m$ of degrees at most $m < n_1$ and to use the orthogonality property in $L^2({S}^2)$.