Automated Restore process is not restoring the associated differential backup. It is only restoring the full backup and ignoring the Differentials and Tlog backup files.
When viewing xp_restore_headeronly, CheckpointLSN is not equal FirstLsn for full backup. DifferentialBaseLSN of Differential and Tlog backups are equal CheckPointLSN of full backup.
Automated Restore compares DifferentialBaseLSN with FirstLSN in the header of the full backup file. These values do not match.
The FirstLSN value is the log sequence number of the first transaction in the backup set. The LastLSN value is the log sequence number of the last transaction in the backup set. The CheckpointLSN value is the log sequence number of the most recent checkpoint. The DifferentialBaseLsn is the log sequence number of the most recent full database backup.
For a differential backup set, the DifferentialBaseLsn value tells us which full database backup is required in order to apply the differential backup. LiteSpeed should look for a full database backup file that has a CheckpointLSN value equal to the differential backup's DifferentialBaseLsn value.
NOTE: If database is idle FirstLSN and CheckpointLSN will be the same. If transaction started before backup CheckpointLSN will contain LSN of a backup checkpoint and it will be greater than FirstLSN.
Issue fixed in version 7.5.3. The latest version of LiteSpeed for SQL Server can be downloaded here.