JAKARTA — Southeast Asia hosts some of the world’s leading coffee growers and exporters, such as Indonesia and Vietnam, the world’s fourth- and second-biggest producers respectively. Neighbors such as East Timor, Laos, Myanmar, the Philippines and Thailand are home to smaller coffee-growing regions. Their beans have typically been for export. Many end up in cups in countries such as the U.S., the biggest importing country, and Italy, the third-biggest importer, according to the United Nations Food and Agriculture Organization. Italy is also the home of espresso, the strong and sometimes bitter caffeine shot that for many drinkers is the quintessential coffee pour. Now, coffee consumption is on the rise in what were mainly exporting countries of the region, and the race is on to see which kind of coffee will lure would-be local drinkers.