Hill Control victory conditions

[Shad]

Does the Italian player need to run around on the hill hexes after eliminating or forcing off the British/Punjab in order to obtain control of ALL hill hexes? 

Strictly taking rules as written, yes, the Italian player would need to walk a unit/units through every single hill hex.

Personally, I only play this way if a hill ends contested - last entered = current occupier. If one side stands alone on the hill I just give it to them.

But that's not the rules as written.