  A conjecture raised in 1990 by C. Huneke predicts that, for a $d$-dimensional
Noetherian local ring $R$, local cohomology modules of finitely generated
$R$-modules have finitely many associated primes. Although counterexamples do
exist, the conjecture has been confirmed in several cases, for instance if
$d\leq 3$, and witnessed some progress in special cases for higher $d$. In this
paper, assuming that $R$ is a factorial domain, we establish the case $d=4$,
and under different additional conditions (in a couple of results) also the
case $d=5$. Finally, when $R$ is regular and contains a field, we apply the
Hartshorne-Lichtenbaum vanishing theorem as a tool to deal with the case $d=6$.
