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I agree there is no grace period for L-1/2.  But I don't think USCIS will aprove c/s from L-2 to F-1 simply because the L-2  is not in lawful status when the school starts in June 20.

-Not really. If he files I-539 before his L-2 expires, normally it will not affect the F-1 approval-this is called timely filing. See:

http://go2immigration.com/change-of-status-denied-and-i-94-expired-what-are-my-options/

http://www2.gtlaw.com/practices/immigration/newsletter/archives/014/item02.htm

 

L-2 was in status when the c/s was filed.  That's not enough.  USCIS will likely issue a RFE and hopefully the L-1 extension is approved at that time, provided L-1 holder has concurrently filed I-539 to extend L-2 status.