  We show that the Szeg\H{o} matrices, associated with Verblunsky coefficients
$\{\alpha_n\}_{n\in\mathbb{Z}_+}$ obeying $\sum_{n = 0}^\infty n^\gamma
|\alpha_n|^2 < \infty$ for some $\gamma \in (0,1)$, are bounded for values $z
\in \partial \mathbb{D}$ outside a set of Hausdorff dimension no more than $1 -
\gamma$. In particular, the singular part of the associated probability measure
on the unit circle is supported by a set of Hausdorff dimension no more than
$1-\gamma$. This proves the OPUC Hausdorff dimension conjecture of Barry Simon
from 2005.
