  We consider the well-studied problem of learning intersections of halfspaces
under the Gaussian distribution in the challenging \emph{agnostic learning}
model. Recent work of Diakonikolas et al. (2021) shows that any Statistical
Query (SQ) algorithm for agnostically learning the class of intersections of
$k$ halfspaces over $\mathbb{R}^n$ to constant excess error either must make
queries of tolerance at most $n^{-\tilde{\Omega}(\sqrt{\log k})}$ or must make
$2^{n^{\Omega(1)}}$ queries. We strengthen this result by improving the
tolerance requirement to $n^{-\tilde{\Omega}(\log k)}$. This lower bound is
essentially best possible since an SQ algorithm of Klivans et al. (2008)
agnostically learns this class to any constant excess error using $n^{O(\log
k)}$ queries of tolerance $n^{-O(\log k)}$. We prove two variants of our lower
bound, each of which combines ingredients from Diakonikolas et al. (2021) with
(an extension of) a different earlier approach for agnostic SQ lower bounds for
the Boolean setting due to Dachman-Soled et al. (2014). Our approach also
yields lower bounds for agnostically SQ learning the class of "convex subspace
juntas" (studied by Vempala, 2010) and the class of sets with bounded Gaussian
surface area; all of these lower bounds are nearly optimal since they
essentially match known upper bounds from Klivans et al. (2008).
