No. The difficulty will not increase. There will only be 12.5 BTC every 10 minutes, period. No matter what the difficulty is.
I presume the difficulty will decrease because many people will not be able to pay for the electricity mining only half of what they are mining now.
Until the price doubles, then they will turn on their rigs again.
Difficulty will not decrease if price stay high or even better continue rising.
But if price falls down miners will definitely be in problems.
They won't sell under the price and the price will correct soon.