Hill Control victory conditions

[tlangston28]

I'm playing Desert Rats scenario #1 using 4th Ed rules and in reading the victory conditions,  hill control is key to Minor and Major victory and reads "...ALL hill hexes are controlled..." (emphasis my own).   Reading RAW,  Control of all the hill hexes in this particular scenario is currently British/Punjab (setup on the hill, enemies entering the board). In addition,  Control is defined by an undemoralized combat unit being the last to occupy a hex exclusively.

The question then is, by these rules,  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?  

I couldn't find the question anywhere else and in reading the AARs it sounds like most of the players were playing to "clear" the hill rather than "control" the hexes on the hill.