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Experts say the first direct talks between the Taliban and Afghan officials are crucial because the Islamists had previously refused to engage with Kabul. The US has hailed substantive progress on all parts of a deal.
Zalmay Khalilzad, the US envoy for Afghanistan, on Saturday said the latest round of talks with the Taliban in Doha, Qatar, has been the "most productive" so far.
"The six days have been the most productive of the rounds we have had with the Taliban… we made progress on all the issues that we have been discussing," Khalilzad said in Doha.
US-Taliban talks will resume on Tuesday after an intra-Afghan dialogue, facilitated by Germany and Qatar.
Suhail Shaheen, the spokesman for the Taliban's office in Qatar, also expressed his satisfaction over the latest round of peace talks with the US.
"We are happy with the progress and hope the rest of the work is also done. We have not faced any obstacles yet," he tweeted.
The earlier rounds of US-Taliban talks did not yield any substantial result, but hopes for a breakthrough are high this time around. The main reason behind this optimism is the Taliban's willingness to engage with the Afghan government, which complained of being sidelined in peace negotiations.
The Islamists, who ruled the country from 1996 until the US invasion of Afghanistan in 2001, had dubbed the Afghan government a "US puppet" and refused to talk to them.
Read more: Taliban-US peace talks reopen in Doha
Afghan President Ashraf Ghani insists that no agreement to end the 18-year conflict in his country could be reached without his government's active involvement.
It appears that the Taliban have agreed to show flexibility on their dealings with Kabul, but the group insists that the Afghan delegates attending the two-day talks will only do so in a "personal capacity."
Washington has made it clear, however, that the Taliban must talk with the Afghan government if they wish to seal a peace deal.
A previously planned meeting between Afghan representatives and the Taliban in April collapsed due to a disagreement over the size of the proposed 250-strong Afghan delegation as well as over its official status.
This time at least 60 Afghan delegates are reportedly taking part in the two-day intra-Afghan dialogue starting on Sunday. The Afghan delegation includes various stakeholders, including former mujahedeen (Islamic warriors) who fought the Soviet Union in the 1980s, as well as former government officials, former ambassadors, civil society representatives and a small number of women.
Asila Wardak, an Afghan Peace Council member, who's attending the intra-Afghan talks in Qatar, told DW that the Sunday discussion was "quite general and not specific."
"We'll have a detailed talk later today and on Monday," said Wardak.
The Afghan delegate said that the Taliban representatives admitted that "they do not want a monopoly on power in Afghanistan."
Read more: Afghanistan: Peace without women's rights?
"Some individuals, who represent Afghanistan, are holding direct talks with the Taliban in Qatar. These individuals will share views about the current situation in Afghanistan with Taliban members," Waheed Omer, an adviser to President Ghani, told a news conference in Kabul on Sunday.
"The Afghan government and its international allies are working on an agreement to be sealed before September. We can have 'serious peace talks' after that," Omer added. "Afghanistan can only achieve lasting peace if an elected government represents the Afghan in peace negotiations."
Withdrawal or ceasefire?
US Secretary of State Mike Pompeo said he hopes for a peace agreement before September 1.
The Trump administration, however, is demanding a ceasefire agreement amid continued Islamist violence in the war-ravaged country.
Last week, the Taliban claimed a truck bomb attack in Kabul that killed six people and wounded 105 civilians, many of them children.
But the Taliban demand a timeline for the withdrawal of US troops from Afghanistan before agreeing to a ceasefire.
Wahid Muzhda, a Kabul-based security analyst, told DW this could be risky. "We don't know if the Taliban will then agree to talk to the Afghan government or demand an interim government that they could become a part of," said Muzhda, adding that participants of the meeting urged all sides to avoid civilian casualties in case a ceasefire agreement was not reached.
"Taliban did not comment on a potential ceasefire agreement but US special representative Zalmay Khalilzad said last night that significant progress was made in this regard," Afghan Peace Council member Wardak said on Sunday.
The US official in Qatar denied reports that Washington was ready to pull out all troops from Afghanistan. The official told The Associated Press on condition of anonymity that the US "definitely did not offer" a withdrawal as part of a peace deal.
The US negotiator said the deal that is being hammered out in Doha with the Taliban is "comprehensive and includes specifics on all four parts including a ceasefire, timeline, participating in intra-Afghan negotiations and counterterrorism assurances."
"We can expect progress if we define progress as movement toward a troop withdrawal deal," Michael Kugelman, a South Asia expert at the Washington-based Woodrow Wilson International Center for Scholars, told DW. Such a deal, Kugelman added, could only be counted as a win for the Taliban, not for the US or the Afghan government.
"That's certainly progress for the Taliban, but not for the US, which has repeatedly said that there can be no agreement until both sides agree on all the issues on the table," he underlined.
Additional reporting by DW's Pashtu-Dari department.