Currently, if the pegasus website is down then the PegasusLite jobs fail as download of worker package from the pegasus site breaks. Pegasus allows for explicit worker package staging that allows users to specify a worker package in the transformation catalog to use

https://pegasus.isi.edu/documentation/user-guide/data-transfers.html#staging-of-worker-package

However, using this does not seem to work when a job runs using containers. Even though the user may have set the following property

pegasus.transfer.worker.package=true
pegasus.transfer.worker.package.autodownload=false