Given the little bit of what I know about the new features in GH 6 verses GH 5, I would not even attempt to downgrade without a test environment. I would contact Atlassian support if you really need to downgrade. I would be more concerned about losing data, if you need to recover Jira from backups.
I would probably spend my time convincing your bosses to leaving things alone.
GreenHopper 5.10.5 was one of the last releases of GreenHopper 5.x, so the differences (featurewise) to the current version are not that big. There was even a thread going on here where someone successfully downgraded from 6.0.1 to 5.10.x so it might be easy.
Anyway, as Norman pointed out in the comments above I would strongly suggest to try this on a test environment. You may want to consider duplicating your JIRA instance just for this purpose.
The process of downgrade would than be pretty easy:
As stated above this might just work out. If you used new features like the "GreenHopper simplified Workflow" (stuff that changes other stuff deep within the JIRA configuration) I'm not sure how this would behave after the downgrade. In an ideal world (must be a parallel universe or so) you than would have to manually re-modify your workflow an all is fine. Not completely shure about this though...
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