Not a war but more like a Riot (wink wink). The Piltover/Zaun conflict mirrors socio-economic class warfare like the early 2010s We Are The 99 percent "eat the rich" situations.
So less a war and more a riot. The powerful ChemBarons and the Mercantile families of Piltover likely do not want open conflict. That's the whole point of Camilles' character. A direct conflict makes no sense as it would only disrupt trade and weaken the twin cities for Noxus to step in.
What's more likely to happen is for Piltover and Zaun citizens to try to kill one another over political problems with each other.
The only question is who would be the initial aggressor
I had a head canon where Jhin was sent by a Ionian cabal looking to be more militarist and confrontational with outside nations to ignite a conflict between Piltovans and Zaunites. That's what is going on in the Awaken cinematic. Camille going after Jhin after he murdered an entire theatre of innocent Piltovans. His initial assignment was probably much smaller in scale, but Jhin must find killing for political power painfully rote and like a rabid dog left off the leash has set off to cause chaos on his own accord.
Piltovans would view Zaunites as murdering savages and try to blame them while the Zaunites have had reason enough to hate "Pilties" up until now. This could boil over into larger scale conflict.