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Some call it a pipe dream. Others say it is a grand self-delusion. In the Biden White House, however, it is called a “moonshot.”

Since early March, the new U.S. administration has let it be known that it hopes the Taliban in Afghanistan will form a transitional regime with the elected government in Kabul leading up to national elections. Despite having very different views on Islam, women, and governance, the two warring sides “must find a path to a political settlement,” says Zalmay Khalilzad, the special U.S. envoy on Afghanistan.

The United States wants the Taliban and Afghan leadership to start these power-sharing negotiations in coming weeks, preferably with Russia, China, Turkey, Iran, and Pakistan on the sidelines for added pressure. And to lay down a marker on how they should go, Secretary of State Antony Blinken told Afghan President Ashraf Ghani in a letter that the current constitution can be an “initial template.”

The difficulty in this moonshot is that Mr. Ghani, like most Afghans, considers the 2004 constitution to be sacrosanct, especially in its protection of women’s rights and other democratic principles. The Taliban, meanwhile, cling to a goal of re-creating the Islamic emirate of their rule from 1996 to 2001, just before the U.S. invaded to strike at Al Qaeda after 9/11.

If these newly focused talks actually start, Afghanistan will be doing what has been elusive for many Muslim countries: reconciling a utopian view of an Islamic community living under religious law with the vision of a society of equal citizens committed to basic rights under elected leaders.

President Joe Biden’s strategy is not new. When he was vice president, President Barack Obama said the U.S. would “join initiatives that reconcile the Afghan people, including the Taliban.” At a local level, many Afghan leaders have found ways to talk to the Taliban with respect, often succeeding in moderating their views or persuading them to leave the group. These dialogues start from an Islamic perspective that finds common ground on religious and democratic values.

Part of the Biden plan is to use the talks to convince the “reconcilable” members of the Taliban to break off and opt for peace. The U.S. has already worked with parts of the Taliban in attacking the forces of Islamic State in Afghanistan. The Taliban also hint at softening their position on educating girls. In recent days, they suggested a three-month “reduction of violence” period. And under a tentative deal with the Trump administration last year, the militant group has not attacked U.S. forces, which number about 3,500.

Under that deal, Mr. Biden has inherited a deadline of a total withdrawal of forces by May 1. He has suggested extending the deadline. Perhaps he hopes his moonshot will launch. Other Muslim countries have created constitutional democracies while finding a popular role for Islam. It may not be a pipe dream to think Afghanistan can do the same.