There is a great temptation to make a Bush/Iraq comment here, but I shall refrain.
I don’t think we will ever find evidence of such a thing after so long. The size of the opposing forces would hardly constitute a squad action today, and how would you know if it was a war or just a neighborly mugging? Unless someone finds a cave painting equivalent of the Bayeaux Tapestry, I think this question will go unaswered.
That painting isn’t out of the question.
And if a dozen people are found dead, with wounds caused by weapons, that’s war enough to suit the subscriber.
That may be tough as we have (IIRC) only found the remains of about three hundred Neandertalers from a range of about ninety thousand years. I would state with a high degree of confidence that we do not have a dozen individuals who ever saw each other, much less waged war.
I also do not recall any evidence that Neandertalers created any representational art which has survived to this late date.
Depends on what you mean by “war” in the sense of deliberate aggression by one organized group of people against another in order to gain some advantage such as territory. However, there is pretty good evidence that Neanderthals killed - and subsequently ate - other Neanderthals.