The response from the FSF (who were prompt and useful) said:
In section 13 of the GPL, it states "the
special requirements of the GNU Affero General Public License, section
13, concerning interaction through a network will apply to the
combination as such." That means that if you are making a work that
includes both AGPL and GPLv3 software, then the AGPL's special
requirement applies to the work as a whole. That means that you have a
combined work up and running on a server, you must offer the code as a
whole to users interacting with that combined work.
and
The AGPL's special requirement doesn't turn on whether the work is
'conveyed,' or whether network interaction is conveyance. Instead, the
license just makes special provisions for the situation where users can
interact with a modified version of the code over a network.
That means that an AGPLed library will still need to be redistributed if changes are made to it and it is used behind a web-service front-end. While it does force the service developer to use a GPL or AGPL license on their front-end and redistribute their source, I think that's a minor problem as it doesn't stop commercial derivatives, it just means that they have to be free.
Also, since the AGPL is effectively the GPL with a modification to its definition of "conveyance" so that networked services can't modify GPLed code without redistributing those changes, the compatibility matrix shows that the main IBBoard Utils can stay as LGPLed.