  A family $\mathcal{F}$ on ground set $\{1,2,\ldots, n\}$ is maximal $k$-wise
intersecting if every collection of $k$ sets in $\mathcal{F}$ has non-empty
intersection, and no other set can be added to $\mathcal{F}$ while maintaining
this property. Erd\H{o}s and Kleitman asked for the minimum size of a maximal
$k$-wise intersecting family. Complementing earlier work of Hendrey, Lund,
Tompkins and Tran, who answered this question for $k=3$ and large even $n$, we
answer it for $k=3$ and large odd $n$. We show that the unique minimum family
is obtained by partitioning the ground set into two sets $A$ and $B$ with
almost equal sizes and taking the family consisting of all the proper supersets
of $A$ and of $B$. A key ingredient of our proof is the stability result by
Ellis and Sudakov about the so-called $2$-generator set systems.
