It's not physically possible. All the water that comes into the hydro plant leaves the plant. What would they do with so much extra water otherwise?
The hydro plant simply collects some of the potential energy of the water. River sources are high, river deltas are low - that's why the water flows. Water is heavy. There is plenty of energy in all that water that it loses as it descends. That energy converts into heat, essentially, and heat is the last thing you want in a desert lake (it facilitates evaporation.)
A dam collects some of the water, allowing it to gather and rise back to some level. Then the water is taken from upper levels of the dam and fed downstream through a turbine which cools the water and produces electric energy. The amount of water in front of the dam is fixed and never changes; it amounts to a one-time borrowing of water when the plant opens. The water is still there, though, and can be released back into the lake whenever desired (if the dam is strong enough.)
Aral Sea is drying up because the water from rivers that feed it was taken away for irrigation purposes. That water was physically removed, not just allowed to pool up. That's a big difference.
So this dam is exclusively for power generation, not irrigation?