Cynfael pretty much elaborated the issues about having a third fairy, but I'll do my share as well.
No, I have no solid proof. But who does? Both stories of the third fairy theory cannot be proven and can only be speculated. If you consider the scholar lore, I believe a third fairy will not happen.
As arcanist, you reclaimed an ancient tablet originated from the Nym civilization and with it a soulstone - Which you quickly snatched before the lalafel did, shame on you! The previous owner of that soulstone has succumb to the tonberry epidemic and the soulstone, with the fairy, has slumbered ever since. That single fairy, named Lily, has two sides: Eos and Selene. Both aspects of the fairy has been discovered prior to Lily recovering her memories and was described in the tablet of the nym. You could conclude that the fairy was already aware of the powers that's already lingering inside of her. If the fact that Lily were actually two seperate fairies and not one, the relation between Lily and Eos and Selene, from the Greek mythology, wouldn't be far-fetched at all. But it isn't. Lily is a single entity that happens to have two different aspects/powers. The lore also never mentioned that the tablet was incomplete.
As a scholar, you never had to do anything to unlock your familiars or their abilities. Selene and Eos has always been there from the moment you became a scholar. Summoners, on the other hand, had to battle the egi versions of specific primals before they were able to harness their powers and had a job skill quest that involves unlocking their hidden abilities: Enkindle. At the end of the scholar lore, Lily's memory has returned and the story came to an end. Helios or another potential familiar was never mentioned in the lore.
With that in mind: Lily (your current fairy) will only have two kind of powers: The power to soothe and shield and the power to silence and strengthen.
Of course, Square-Enix could easily create some cheap lore that would change the current lore with a simple "oh we found another tablet" or "hey Lily has more memories undiscovered!", but will they really?
Heck, if Lily had two aspects called Goku and Vegeta, would you expect they'd add Gohan too? The names happen to be inspired by Eos and Selene from the Greek mythology.
Lore aside, the biggest question would be whether a scholar needs another fairy. Both Eos and Selene already do their jobs perfectly well. A third fairy would either be useless or make one or the other obsolete - As mentioned by Cynfael. I believe it's more realistic to expect an ability similar to Enkindle: Teaching the fairy a fifth ability.
And fairy aside, I'd much rather see some really useful abilities and/or spells that would expand/enrich the current Scholar kit. Rather than a fairy, or familiar for that matter, that would completely replace one fairy and only have minor effect.