Your question is the lower part of Oppermann's conjecture that I think to have proved in a publication available at:
http://www.scienpress.com/journal_focus.asp?main_id=60&Sub_id=IV&Issue=1263
and for which I still wait some commentaries.
The title of this article is 'From Sierpinski's conjecture to Legendre'. It has been published in the journal Theoretical Mathematics & Applications, vol.4, no.2, 2014, 65-77, ISSN: 1792-9687 (print), 1792-9709 (online) Scienpress Ltd, 2014. In this article, I show that Sierpinski's, Oppermann's and Legendre's conjectures (see Wikipedia for definitions), all dealing with N², can be integrated into a new global D conjecture which possesses a property of recurrence.This property is used to prove these three linked conjectures respecting the order: Oppermann's first and conditional to D, Sierpinski's second and conditional to D, D third and unconditional, Oppermann's and Sierpinski's fourth and unconditional, and finally Legendre's.
Regards.