They claim that the modern higher efficiency furnaces emit exhaust at a lower temperature than the older furnaces do. On a long vertical run (in this case, something on the order of 25-35 feet or so), the exhaust cools enough that - before it can exit the flue - it hits the relatively colder air toward the top of the chimney and causes condensation.
The claimed fix is to either insulate the flue itself with material made for this purpose, OR to reduce the inner dimension of the chimney to reduce the amount of free air that surrounds the furnace flue.
This seems like nonsense to me, but what do I know. The chimney in question does have a cap on it, though the critter screen that the top is probably compromised. So, I don't think it is direct precipitation getting in. Moreover, the bottom of the cleanout manifests dampness not rivers of water.
Many thanks to all taking the time to comment here ...