Would this pump water up? and if so, how far?

Would this pump water up? and if so, how far?

I had this idea of an osmotic pump way back in high school and I never got
a satisfactory answer if it would work. If I had this configuration:

Would it continually pump water up given ambient heat so long as the
bottom reservoir is full?
EDIT To explain what is happening, there is pure water in each of the dark
blue reservoirs, saline in the cyan containers (same concentration of
saline in each container) and a semipermeable membrane at the bottom and
near the top of each container.
The pure water in each reservoir would be sucked up into each container
directly above it due to osmotic pressure (high water concentration flows
to lower water concentration), and then dumped out at the top also due to
osmotic pressure (saline to air which is almost 0% water concentration).
Since the membrane is not permeable to salt, only the water is released
from the container into the next higher reservoir. END EDIT
If enough were stacked, would this allow for transporting water higher
than the maximum that trees can transport (about 138m according to this
article)?