Source: Asymco.com

FORTUNE — Reuters reports — and the Wall Street Journal confirms — that Apple’s AAPL Tim Cook and Google’s GOOG Larry Page have been talking to each other about patents. According to Reuters, the two have been conducting “behind the scenes” talks about a range of intellectual property issues, and spoke on the telephone last week. According to the Journal, they have actually “met to discuss the intellectual-property disputes.”

The news, predictably, set off a wave of speculation in the tech press about whether the billion dollar Apple-Samsung verdict last week was enough to get the two tech giants to give peace a chance, bury the hatchet, kiss and make up, play nice, negotiate nuclear disarmament, and achieve peace in our time.

That’s a lot to ask of a phone call, especially when the stakes are so high.

Smartphones are the fastest growing segment of the trillion dollar mobile phone industry, thanks in large part to innovations introduced five years ago by the iPhone. To get a feel for what Google’s Android operating system has cost Apple in terms of smartphone market share, you only have to glance at the chart above, released two weeks ago by Asymco‘s Horace Dediu.

Having achieved dominance in a market that generates hundreds of billions of dollars a year, Page is not about to cave over a $1.05 billion verdict against one of Google’s clients. Nor is Cook likely to back off his insistence that competitors “invent their own IP” on the strength of one verdict and a handful of patents. Apple has tens of thousands of patents and dozens of lawsuits pending against manufacturers of Android devices.

One company it hasn’t sued over Android is Google. At least not yet.

See also: Why Apple sued Samsung and Will Apple now sue Google?