Napkins in a Random Seating Order

At a large circular banquet table with $n$ seats, there is one napkin between each pair of neighboring seats. $n$ diners arrive one at a time and fill the seats in uniformly random order.

When a diner sits:

• if both neighboring napkins remain, she chooses left or right uniformly at random;
• if only one remains, she takes that one;
• if neither remains, she gets no napkin.

As $n\to\infty$, what fraction of diners are expected to end up without a napkin?