Where future rows don't exist
MySQL-5.5

Bummer.

It appears Analytical Functions are not supported in v5.5

If it did, the solution would be something like:

select TaskJournalId, TaskId, TaskStatus, TaskStart, TaskEnd, TaskInfo
from (
  select  TaskJournalId, TaskId, TaskStatus, TaskStart, TaskEnd, TaskInfo
    ,row_number() over (partition by TaskJournalId,TaskId order by TaskStart DESC) rn
  from TaskJournal
)
where TaskStatus=3 and rn=1;


My Answer

Upgrade your database.