I very much doubt it. €10 isnt a steep price at all. People that want to play this game wouldn't mind paying for it. And there's a free trial for people who are interested in how things are right now.
It's a 10 year old game. People that wanted to play it are playing it right now, or already have. It's not going to draw in any new players. At most some returnee's who miss the game. They either quit after a month again, or dont mind paying the monthly fee.
And to take it a step further, Free-To-Play would kill off both this game and all of SE's income. With as big of a flop as XIII and XIV were, XI is the only thing making money for them right now. If they were to turn it Free-To-Play (which i can asure you they will not), the last shred of skill and effort doesnt matter anymore. Just who has the biggest RL wallet to buy the largest amount of endgame gear or gil. Yes, income will be generated for SE for a little bit, but i cant think of anyone who would want to keep playing a game where you can just buy yourself into level 99 and relics.
That said, SE absolutely sucks at advertisement these days. I remember the last Square Enix advertisement i saw was for Final Fantasy 7... 15 years ago... It looked so good, it instantly made me buy it. They just havent been trying ever since. If they would have promoted FFXI 10 years ago, instead of relying on a "hype" that would "sell itself", we would have had a 10-fold amount of players. Same for FFXIV (despite a too early release, for a too poor product, that they ruined update after update after releasing it), never saw a single advertisement for it in Europe.