Some fast convergent series for the mathematical constants $\zeta (4)$ and $\zeta (5)$

Recently, Sun [preprint, arXiv: 2210.07238v7] proposed two conjectural series for the mathematical constant $\zeta (4)$ and two conjectural series for the mathematical constant $\zeta (5)$. In terms of the operator method and two hypergeometric transformations, we prove these four conjectures. Furthermore, we also find some new series for the two constants in this paper.