Ellis and the third author showed, verifying a conjecture of Frankl, that any
3
3
-wise intersecting family of subsets of
{
1
,
2
,
…
,
n
}
\{1,2,\dots ,n\}
admitting a transitive automorphism group has cardinality
o
(
2
n
)
o(2^n)
, while a construction of Frankl demonstrates that the same conclusion need not hold under the weaker constraint of being regular. Answering a question of Cameron, Frankl, and Kantor from 1989, we show that the restriction of admitting a transitive automorphism group may be relaxed significantly: we prove that any
3
3
-wise intersecting family of subsets of
{
1
,
2
,
…
,
n
}
\{1,2,\dots ,n\}
that is regular and increasing has cardinality
o
(
2
n
)
o(2^n)
.