The Taliban denied Thursday that the fighters responsible for downing a US helicopter, killing 38 people, had been killed. The comments came after General John Allen, commander of the NATO-led foreign force in Afghanistan, said Wednesday that the insurgents responsible had been hunted down and killed in an air strike. But Taliban spokesman Zabiullah Mujahid told AFP: "This is not true. After seeing the enemy statement, we contacted the mujahed (fighter) who shot down the helicopter and he's not dead. He's busy conducting jihad elsewhere in the country." Mujahid added that four Taliban fighters had been killed in the US air strike but not those who shot down the helicopter. The Taliban is known to exaggerate and distort its public statements in relation to the insurgency. Allen, however, said the insurgent leader who was the target of the original operation in which the Americans perished remained at large. "Did we get the leader that we were going after in the initial operation? No, we did not," he told reporters by video link from Kabul. In the deadliest incident of the war for NATO, insurgents shot down the Chinook on Friday in eastern Afghanistan, killing 30 US troops, including 25 elite special forces, as well as seven Afghan commandos and an interpreter. Allen said that "at approximately midnight on 8 August, coalition forces killed the Taliban insurgents responsible for this attack" in a bombing raid by an F-16 fighter jet. The general called the downing of the chopper in the remote Tangi Valley a "tragic incident" but portrayed the retaliatory strike against the insurgents as proof that the United States would press ahead with the war. "This does not ease our loss, but we must and we will continue to relentlessly pursue the enemy," said Allen. The helicopter attack came at a sensitive moment as US forces begin a gradual withdrawal and amid growing anxiety in Congress about the cost of the conflict that has dragged on since 2001. Allen announced the air strike against the insurgents amid criticism of the Pentagon's handling of the crash. The ceremony for the return of the remains of the fallen troops, attended by President Barack Obama and other top officials, was closed to the media and the names of those killed have yet to be released -- in a break with Defense Department practice. But Pentagon spokesman Colonel Dave Lapan said Wednesday the identities would be released within 24 hours, after Defense Secretary Leon Panetta overruled objections from officers at US Special Operations Command. Describing the helicopter crash in detail for the first time, Allen said the Chinook had been sent in as part of an operation targeting a Taliban leader. "The intelligence that had been generated to this point led us to believe there was an enemy network in the Tangi Valley in the Wardak province, and the purpose of this mission was to go after the leadership of that network," the general said. When "less than 10" insurgent fighters were seen "escaping," the Chinook chopper carrying 22 Navy SEAL commandos, three Air Force special forces and Afghan soldiers was ordered in to head them off, he said. The CH-47 was then shot out of the sky with a rocket-propelled grenade, killing all 38 people on board. Afterward, US forces tracked the insurgents responsible, calling in an air strike late Monday with an F-16 fighter. The insurgents were traced over the weekend to a wooded area in Chak District "after an exhaustive manhunt" by Special Operations Forces, the NATO-led International Security Assistance Force said in a statement. The airstrike killed the "shooter" as well as a Taliban militant, Mullah Mohibullah, as they "were attempting to flee the country in order to avoid capture," ISAF said. Asked if he had concerns about the use of the larger, slower Chinook in Friday's operation or the deployment of such a large number of elite special forces as a quick reaction force, Allen said he was "comfortable" with the decisions made. "We've run more than a couple of thousand of these night operations over the last year, and this is the only occasion where this has occurred," he said, adding that the CH-47 has been routinely deployed.