Gale's subset take-away conjecture

Let $A$ be a finite set. In subset take-away, two players alternately choose proper, non-empty subsets of $A$, with no chosen set containing a set chosen earlier; a player unable to move loses. Gale's subset take-away conjecture. Subset take-away is always a second player win. This conjecture, attributed to David Gale, asserts that the second player has a winning strategy for every finite starting set. The paper verifies the claim for starting sets with fewer than six elements and presents it as open in general.