Taliban militants headed into cities across Afghanistan on Sunday as they celebrate their unprecedented Eid ceasefire, raising questions about what happens should the hugs and selfies stop and the festive atmosphere and calls for peace turn sour. President Ashraf Ghani extended the government ceasefire on Saturday, garnering praise at home and international backing, but critics said such overtures have allowed the Taliban to pour into cities unchecked and win the upper hand. Ecstatic men and children crowded around the soldiers and Taliban fighters, some who had checked in their weapons at the entrances to cities, over the weekend and urged them to extend their ceasefire into a permanent peace. Governors and senior government officials hosted small feasts, played music to welcome the Taliban after they announced a three-day ceasefire over Eid, marking the close of the month-long Ramadan fasting season that ends on Sunday. Ghani, in an address to the nation, urged the Taliban to extend their ceasefire and begin peace talks. He said he wanted to discuss “issues of mutual concern with neighboring countries and the presence of foreign forces”. His office has yet to declare a new timeframe for the extended ceasefire. A senior western diplomat in Kabul said Ghani's decision was “a bold move” but questioned what would happen if the Taliban rejected the fresh ceasefire request. “The consequences could be disastrous,” he said. Members of Afghanistan's High Peace Council supported the decision and urged the Taliban to extend their ceasefire. “Good steps for peace have been taken, but more steps have to be taken,” said council chairman Karim Khalili. U.S. Secretary of State Mike Pompeo, echoing Ghani's remarks, said peace talks would have to include a discussion on the role of “international actors and forces”. In August, U.S. President Donald Trump unveiled a more hawkish military approach to Afghanistan, including a surge in air strikes, aimed at forcing the Taliban to the negotiating table. “Ghani has full blessings of the U.S. administration and it is crucial for the U.S. officials to prove that Trump's policies are working and talks with the Afghan Taliban are imminent,” said a senior diplomat who met Afghan officials on Sunday to monitor the chance of back-channel talks with Taliban leaders. The United States wants Pakistan, which it accuses of harboring Afghan Taliban commanders, to exert more influence on the group to bring it to the negotiating table. The only time direct talks between the Afghan government and the Taliban have been held, in 2015, they broke down almost immediately. The Taliban are fighting U.S.-led NATO forces, combined under the Resolute Support mission, and Ghani's U.S.-backed government to restore sharia, or Islamic law, after their ouster by U.S.-led forces in 2001. But Afghanistan has been at war for four decades, ever since the Soviet invasion in 1979. A car bomb killed 36 people at a gathering of Taliban and Afghan armed forces in the eastern city of Nangarhar on Saturday. Islamic State claimed responsibility. On Sunday, Taliban insurgents were traveling freely with their flags and meeting civilians despite curbs imposed on their movement after Islamic State militants targeted the gathering. “In order to prevent our countrymen from being harmed and... we become a reason for that, all officials should ban all mujahideen (Taliban fighters) under their command from participating in such crowds and gatherings,” said Zabihullah Mujahid, the Taliban spokesman. Ghani had committed “a grave mistake” by allowing Taliban fighters to enter government-controlled areas, said Amarullah Saleh, a politician and a former head of the National Directorate of Security (NDS). “We don't have mechanisms in place to mitigate the breach of ceasefire by the Taliban,” Saleh told Reuters. Members of parliament opposing Ghani's move said he had not consulted politicians and would be left with no recourse if the Taliban rejected his impromptu request. The Taliban already roam huge swaths of the country and, with foreign troop levels of about 15,600, down from 140,000 in 2014, there appears little hope of outright government victory. For now the atmosphere is festive - a story doing the rounds on social media is that one Talib, after hugging Kabul residents, asked for directions to Baharistan, to the west of Kabul. “I have heard there is very good ice cream there,” he said.