  A clutter is \emph{$k$-wise intersecting} if every $k$ members have a common
element, yet no element belongs to all members. We conjecture that, for some
integer $k\geq 4$, every $k$-wise intersecting clutter is non-ideal. As
evidence for our conjecture, we prove it for $k=4$ for the class of binary
clutters. Two key ingredients for our proof are Jaeger's $8$-flow theorem for
graphs, and Seymour's characterization of the binary matroids with the sums of
circuits property. As further evidence for our conjecture, we also note that it
follows from an unpublished conjecture of Seymour from 1975. We also discuss
connections to the chromatic number of a clutter, projective geometries over
the two-element field, uniform cycle covers in graphs, and quarter-integral
packings of value two in ideal clutters.
