Kashina's exponent divisibility conjecture for semisimple Hopf algebras

Let $H$ be a semisimple Hopf algebra over $\mathbb{C}$. Kashina's conjecture. The exponent of $H$ divides $\operatorname{dim}(H)$. This conjecture is a divisibility analogue of the classical result that the exponent of a finite group divides its order; the source identifies it as a conjecture considered by Kashina, but the supplied text does not establish its resolution.