One LPAR loses connectivity to a structure, and a queue manager on another LPAR has connectivity
SYSTEM MANAGED REBUILD is initiated by the queue manager to move the structure to another CF (if one with better connectivity is available).
In this case persistent and non-persistent messages will be kept.
All LPAR lose connectivity to a structure
If all LPARs lose connectivity to the structure then the structure is recreated, possibly in an alterative CF (your structure CFRM PREFLIST attribute must contain multiple CFs) and persistent messages are recreated by reading the log for the last CF backup, and merging updates since the backup.
Non persistent messages will be lost.
The logs from all queue managers that have accessed the structure since the last backup will be required.
If you do not have the logs to be able to recover,. then the command RECOVER CFSTRUCT TYPE(PURGE) will delete all messages in the structure, and make the structure available for use. You will lose any messages you had in this structure.
If you cannot recover the structure CSQSYSAPPL, you should contact IBM because the recover cfstruct type(purge) command will not work. This structure has information about the status of units of work, and so purging the content on this structure may cause data integrity issues from applications.
If you lose connectivity to the admin structure this will be recreated, possibly on a different structure and the queue manager will rebuild the data from its saved state information, which may require logs.