SLANGERUP, Denmark—For more than a century, businesses have struggled to solve a curiously complicated challenge: How to make a paper bottle that doesn’t get soggy and keeps drinks fresh. Now they say they are the closest they have ever been. Diageo, Pernod Ricard and Procter & Gamble are among a raft of companies testing paper-bottle designs they are betting can help their brands stand out on shelves, woo consumers concerned about plastic and cut carbon emissions. …The paper-bottle push comes as paper is growing in popularity as a substitute for plastic packaging, with companies already using it to sell chocolate, ice cream, chewing gum and chips. …Environmentalists have questioned the merits of substituting paper packaging for plastic. …Despite the uncertainties, consumer-products companies are plowing ahead. Their holy grail is a paper bottle that is easy to recycle, avoids fossil fuel-based plastic and—ultimately—boosts sales. [to access the full story a WSJ subscription is required]