Why he pulled nose-up is a mystery. Maybe he thought the 2 brief stall warnings were false, maybe he wanted to try and climb above the icing or the turbulence. However he did not have reliable airspeed indication in these moments, to attempt this.
As I read the report, it appears that the stall warnings were as a result of him pulling the nose up, not what caused him to the pull-up
Could he have felt the floor fall out from under him as a result of turbulence, and as a reflex reaction thought he needed to pull the nose up to maintain altitude?