Just clarifying @PeterHumphries' comment: the product $H_{n_2}H_{n_3}$ is certainly homogeneous, and by one of the set-up lemmas in an elementary treatment of spherical harmonics, can be written as a sum of terms of the form $(r^2)^k\cdot f_{n_2+n_3-2k}$ where $f_{n_2+n_3=2k}$ is a homogeneous and harmonic polynomial of degree equal to its subscript. Integrals on the sphere ignore the powers of radius, of course, so orthogonality of harmonic polynomials of different degrees still does give the result, since $$ n_2+n_3-2k \le n_2+n_3 < n_1 $$

Just clarifying @PeterHumphries' comment: the product $H_{n_2}H_{n_3}$ is certainly homogeneous, and by one of the set-up lemmas in an elementary treatment of spherical harmonics, can be written as a sum of terms of the form $(r^2)^k\cdot f_{n_2+n_3-2k}$ where $f_{n_2+n_3-2k}$ is a homogeneous and harmonic polynomial of degree equal to its subscript. Integrals on the sphere ignore the powers of radius, of course, so orthogonality of harmonic polynomials of different degrees still does give the result, since $$ n_2+n_3-2k \le n_2+n_3 < n_1 $$