Midlevels Conjecture

[N. J. A. Sloane]

In Peter Winkler's very nice new book "Math Puzzles"
he mentions an unsolved problem, which is related to a
possibly new sequence:

a(n) = number of ways to cycle through all the
subsets of size n and n+1 of a set of size 2n+1 by adding or
deleting one element at a time

a(1) is 2, because the two cycles are

1 13 3 23 2 12 1 ...

and its reverse

The conjecture is that a(n) is always > 0.

Could some seqfan work out a few terms?

Neil