On Monday they announced the release of scores of detainees they had rounded up and held for years in rebel-controlled territory.
The abovementioned deal makes part of UN Security Council resolution calling for a 30-day deployment of a United Nations team to Yemen to help monitor the ceasefire in Al Hudaydah during the UN-led consultations in Sweden.
Yemen's interior minister earlier denied that the incident had taken place and said video released by the Houthis was old footage. He expressed hope this would open the door to "further releases to bring comfort to families awaiting reunification with their loved ones".
Yemen's war has killed tens of thousands of people and sparked the world's worst humanitarian crisis.
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While Yemen's Iranian-allied Houthi rebels claimed the assault, Saudi Arabia says it was "unquestionably sponsored by Iran". The crude oil rates saw on of its biggest single-day jump on September 16 after Saudi Arabia drone attack.
The conflict escalated the following year when Saudi Arabia and its Sunni-Arab allies launched a massive air campaign aimed at rolling back Houthi gains in Yemen and supporting the country's pro-Saudi government. The fighting in the Arab world's poorest country has also left millions suffering from food and medical care shortages and pushed the country to the brink of starvation.
The Houthi National Committee For Prisoners' Affairs said the release was made as a statement of goodwill.
He also claimed hundreds of Saudi soldiers lay dead or injured on the battlefield, and Riyadh had little option but to consider how to withdraw. Each side was meant to release around 7,000 prisoners.
Yahia Sarie, a spokesman for Houthi forces, claimed in a news conference Sunday the rebels took captive more than 2,000 troops, without offering evidence.
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President Donald Trump warned the USA was " locked and loaded " to respond to the attacks, as Breitbart News reported . That moment aside, the Trump administration has seemed willing-and eager, at times-to start a war with Iran.
However, he described Monday's release of prisoners as created to "break the deadlock that has prevailed for several months".
The UN Special Envoy to Yemen Martin Griffiths has arrived in Sanaa on an unannounced visit, a source at the city's airport revealed.
For his part, United Nations special envoy for Yemen Martin Griffiths hailed the move, saying it would ease prisoner swaps between the warring sides within the scope of the cease-fire deal concluded in Sweden. But the deal was never fully implemented. UK, Australia, Bahrain, Saudi Arabia and UAE have so far joined the US-led coalition.
When asked about the prisoner release, al-Malki said he did not have any more information on it.
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Asked about prospects of mediatory efforts of other countries between Iran and Saudi Arabia, Mousavi said in his Monday presser that "The possibility of settling differences through dialogue has always been there".