The linear independence of $1$, $ζ(3)$, and $ζ(4)$

In a recent breakthrough, Fauzan proved the irrationality of $ζ(5)$. In this paper, we combine Fauzan's determinant method with classical simultaneous approximations of zeta values studied by Zudilin and Dauguet--Zudilin to prove the linear independence of $1$, $ζ(3)$, and $ζ(4)$ over $\mathbb{Q}$.