Complementary-response conjecture for subset take-away

Let $A$ be a finite set, and consider the subset take-away game in which players alternately choose proper, non-empty subsets of $A$, with no chosen set containing a set chosen earlier; a player unable to move loses. For an opening move $S\subsetneq A$, let $A\setminus S$ be its complementary move. Complementary-response conjecture. A winning response to an opening move is to play the complementary move. The paper reports computer verification of this response for the relevant case when $|A|=6$, after Gale's conjecture had been verified for smaller starting positions, but gives no resolution of the general conjecture.