The indicator function of the following 96-element set is a counter-example:
$$ \{ 0, 1, 02, 12, 03, 13, 023, 123, 04, 14, 024, 124, 034, 134, 0234, 1234, 05, 15, 025, 125, 035, 135, 0235, 1235, 045, 145, 0245, 1245, 0345, 1345, 02345, 12345, 06, 16, 026, 126, 036, 136, 0236, 1236, 046, 146, 0246, 1246, 0346, 1346, 02346, 12346, 056, 156, 0256, 1256, 0356, 1356, 02356, 12356, 0456, 1456, 02456, 12456, 03456, 13456, 023456, 123456, 7, 27, 37, 237, 47, 247, 347, 2347, 57, 257, 357, 2357, 457, 2457, 3457, 23457, 67, 267, 367, 2367, 467, 2467, 3467, 23467, 567, 2567, 3567, 23567, 4567, 24567, 34567, 234567 \} $$
where (for instance) $2467$ is shorthand for the set $\{2, 4, 6, 7\}$.