Sm1smom
Super Moderator
I would like to add to the answer that mom gave you, that if the background check will stay pending for more than 180 days and the case is otherwise approvable, USCIS is forced to process the case and approve you. Of course if later on your background check will result to have some issues, you risk that your Green Card will be revoked. But at least you won't go over the deadline of the end of the FY. Since in September 30 your background check would be far over the 180 days, I think you are in the safe side. In case your background check will still be pending in June/July though, I suggest you to ask help to one of the senator of your State. They are pretty good in those situations.
The reference of what I said about the 180 days of pending background check is here (search 180 days in the page): https://www.uscis.gov/archive/archive-news/questions-and-answers-background-check-policy-update
Before everyone starts shouting "eureka", I would caution against totally relying on the information on that link. The 2008 memo referenced in that archived Q/A has actually since been superseded by another one, USCIS is not 'forced' to approve a case simply because the background check has been pending 180+ days.
A Feb. 2009 memo has since revised the automatic approval of all such applications after 180 days. Such cases cannot be approved by an applicable FO without prior authorization from USCIS headquarters, the new memo actually talks about contacting the headquarters when the case has been pending for 150 days plus, however that process is not as straightforward as it sounds - a case by case specific guidance will be provided by USCIS headquarters.
2009 Memo:
http://www.legalactioncenter.org/sites/default/files/docs/lac/Feb92009namecheckmemo.pdf
More guidance regarding that memo:
http://www.immigrationpolicy.org/clearinghouse/litigation-issue-pages/mandamus