Hey guys,
After posting this I realised there was a way to remove any Prototype
reference, which made it clear that it's some issue with Firefox.
A test page with the simplified code is at
http://advancedcontrols.com.au/parent.html
. FF displays "false", while Google Chrome displays "true".
Anyway, it's useful to have this issue archived here, particularly the
fact that including Prototype in the child is a work-around.