  We prove that on a general hypersurface in $\mathbb{P}^N$ of degree $d$ and
dimension at least $2$, if an arithmetically Cohen-Macaulay (ACM) bundle $E$
and its dual have small regularity, then any non-trivial Hodge class in
$H^{n}(X, E\otimes\Omega^n_X)$, $n = \lfloor\frac{N-1}{2}\rfloor$, produces a
trivial direct summand of $E$. As a consequence, we prove that there is no
universal Ulrich bundle on the family of smooth hypersurfaces of degree $d\geq
3$ and dimension at least $4$. This last statement may be viewed as a
Franchetta-type conjecture for Ulrich bundles on smooth hypersurfaces.
