Taking advantage of a warm afternoon, I opened the hive and... sure enough, the bees are dead.
They had taken all the syrup, so the bees were definitely in good health at the end of the year. My suspicion is still damp. Next year, I'll use a crown board with two holes. As soon as I can, I'll be taking it apart, cleaning it (bicarbonate of soda, apparently) and starting all over again.