Cannot file COE before graduation?

Hey all, I would like to know if anyone here is having or had same situation with me and could share some information.
I recently got an 12-week internship offer from a US company in Japan, and I am currently processing the COE and visa needed for the internship. I am a University senior student graduating in May and will start internship during summer, from June to Aug. After the internship I will attend graduate school. For the paperwork, I submitted a document with "expected graduation date" to the immigration service of the company, but they told me the COE can only be filed after graduation. I only have 14 weeks for summer and I'm afraid there will not be enough time for me to process a COE, get a visa and then start internship.
I've read from some Japanese website saying that international students in Japan can file a COE before graduation and pick up the COE once they graduated. But I don't know if the information is up to date or can be applied to my case.
I wat to know if it is Really impossible to file COE before graduation... Will appreciate any information!!!

Immigration told you all you need to know; nothing to add.

Actually, it's the company that told you that. You could ask the immigration bureau directly for confirmation (contact information on their website), but either way, at the end of the day it's the company that will submit your CoE application on your behalf, so if they refuse to do it until you have graduated, there is nothing you can do, even if the immigration bureau would accept it.