Does water maintain equal level in connected vessels? Consider a system as illustrated below:

There are two 55 gallon barrels used to store water coming down the gutter.
These barrels are connected together on the bottom to maintain equal water level between them.
They are fed by a 4" vertical pipe connected to a 2" horizontal pipe.
When the 4" vertical pipe on one end fills with water, the water will run into the horizontal pipe and begin to fill the barrels. 
Will the barrels maintain the same level as in the 4" dia. vertical pipe, assuming input flow rate isn't too much to overwhelm the 2" horizontal pipe?
In other words: is $h_1 = h_2$ in the diagram?
I think it should be so, so if it is true what physical principle is at work to keep the levels equal when the diameter of the barrel is different than the diameter of the pipe?
 A: Yes, the levels will all be at the same height.
The pressure (relative to atmospheric) of a water column open at top to the atmosphere is linearly proportional to the depth at that point from the surface of the water.  Very roughly, the pressure goes up about 1 PSI for every 2 feet down from the surface.  Imagine if one barrel had 1 foot of water in it and the other 3 feet.  The pressure at the bottom of the second barrel would be 1 PSI higher than at the bottom of the first.  This would cause water to flow thru the horizontal pipe from the second to the first.
The only way this system is stable (water isn't actively flowing around changing levels in the barrels), is when the pressures at the bottom of the two barrels are the same, which means the heights of the water in each barrel are the same.  All the water surfaces are at the same height, so all the pressures everywhere at the same height are the same.  When this is not the case, water will flow until it is the case.
A: If the barrels are not open at the top the air inside will create a barrier but the air in there can be compressed so if water comes down the gutter pipe it will create more force upon the water thus compressing the air in the tanks. At some point there will not be enough force to overcome the pressure in the tank and it will either equalize or return force it back up the gutter. 
A: If the barrels are sealed, there is no requirement that the levels be equal. They can be both lower or higher, depending on the air pressure at the top of the barrel.
As a simple demonstration, fill a bucket with water, sink a glass into it, and then raise it upside down: the water level will rise above that of the bucket, even if there is air trapped at the top, because the pressure at the top of the glass is much smaller than atmospheric pressure.
For something a bit more extreme, try this:

Image source
A: This happens because (according to Pascal's law of transmission of pressure in liquids) pressure exerted in an enclosed, incompressible fluid is transmitted equally and undiminished in all directions throughout the fluid
A: Assuming the barrels are open to atmospheric pressure, the simple and obvious answer is the levels must even or just as they would if granular material were placed in the containers -  ignoring friction, as the weight of the liquid in the higher-filled containers pushes with more force than the others and that cannot remain in static equilibrium so the result is the even out (thus a function of mass not pressure).
A: If all 3 are open to the same atmospheric pressure the only thing that would alter the levels (temporarily) is the kinetic energy of the water moving through the connecting pipes. It will push the level slightly higher in the container it is moving towards and pull the level down in the one behind. Once the kinetic energy is spent due to friction and increasing pressure resistance in the higher level, then the process will reverse, as the now higher level will flow back to the lower level. This cycle will continue for a short while until kinetic energy is lost to friction, then the levels will be the same. Similar to a pendulum moving back and forth, but a pendulum will move more cycles because of much less friction.
A: it depends upon the dimension of two barrels ,open to atmosphere. if both have same dimension then using bernoulli equation both will have same level of water otherwise not
