British Foreign Secretary Dominic Raab called on the leader of the Houthis to free a Briton held illegally for four years in a Yemeni prison.
Luke Symons, from Cardiff, Wales, was detained in April 2017 in Sanaa, accused of spying for the UK, but has never been put on trial.
Mr Raab wrote to Abdulmalik Al Houthi, the head of the Houthi movement that controls Yemen's capital, to urge him to release the 29-year old immediately and offered to help get the Muslim convert, his Yemeni wife and their young son to the UK.
Mr Symons' family was told that the foreign secretary wrote that several hundred wounded Houthi fighters were returned to Sanaa on two planes after medical treatment last year, but there had been no words on Mr Symons' release despite expectations of a concession.
The return of the fighters in October appeared to be connected to the release of two Americans in a deal brokered by Saudi Arabia and Oman.
The push for Mr Symons' release came as Saudi Arabia opened an initiative to end the six-year conflict with a ceasefire, the reopening of Sanaa's airport and efforts to rebuild the country.
The Houthis control much of the country, including Sanaa, and rebuffed the Saudi overtures which had support from the US, the UN and the internationally recognised Yemeni government.
Mr Symons, known as Jamal after converting to Islam, travelled to Yemen before the war that left the country in the grip of the world's worst humanitarian crisis.
He made several failed attempts to leave Yemen with his family before he was arrested at a checkpoint in Houthi-held Sanaa.
His family believe he was detained only because security forces discovered he had a British passport. His wife and son, 4, are in the south-western city of Taez and have made sporadic visits, including one last week.
"The young man is not happy," said Robert Cummings, his grandfather, who leads the campaign for his freedom.
“He’s asking when he is going to be freed – a question that we can’t answer.
“Nobody from the British government has stood up and asked the question: Why is Luke still in prison? He has never been charged, never been sentenced. Why is he there?”
He has been seen once by a UN official after being taken from prison to a nearby building. The meeting was monitored by security guards and he later told his family he was unable to give details about his treatment in jail.
Human rights groups report widespread torture in Houthi-run prisons.
He stayed behind bars despite a Houthi investigation finding no evidence that Mr Symons was involved in espionage.
Release papers were signed in December 2018 at a time when UN efforts to end the conflict appeared to be making progress with an agreement in Stockholm, Sweden, for a series of ceasefires and a prisoner exchange.
But rivalries between Houthi factions blocked the release plans and the peace proposals stalled.
Hopes were raised again last year when Mr Symons was moved from a political prison in Sanaa to another jail in the city. His family hoped the move was the final preparation for his release but it proved to be another false dawn.
Mr Cummings said he was angry that the UK government had failed to get his grandson released despite Britain being a major aid donor to Yemen.
The UK has given more than £1 billion ($1.36bn) in aid since the start of the war in March 2015. It also spent £342,000 in direct support to the peace process and seconded advisers to UN special envoy Martin Griffiths, the government said.
The UK's Foreign Ministry declined to comment on any letters sent to the Houthi leadership.
"We appreciate this is a difficult time for Luke Symons and his family," a representative said.
"Our staff have been working intensively, and are engaging at all levels with regional partners, to secure his release."
- 2012: Luke Symons, from Cardiff, Wales, travels to Saudi Arabia before moving to Yemen where he meets his future wife in 2014.
- March 2015: Yemen's President Abdrabu Mansur Hadi is forced to flee after an offensive by the Iran-backed Houthi rebels.
- August 2015: Coalition ground troops land in the southern port city of Aden.
- November 2016: The couple have a baby son.
- April 2017: Mr Symons is detained and accused of being a British spy after he is stopped at a checkpoint in the capital Sanaa.
- November 2018: Yemeni official who claims to be scrutinising Mr Symons' file says he will be released in the "coming days". Nothing happens.
- December 2018: Senior member of Houthi decision-making body demands the release of Mr Symons – but local officials ignore the order.
- March 2019: Jeremy Hunt, then UK foreign secretary, raises the case with Yemeni authorities on a trip to the Gulf.
- April 2019: An official for the UN special envoy for Yemen meets Mr Symons and reports he is in good health.
- May 2019: The family speak out for the first time over their frustration at the failures of the UK and UN to secure his release.
- March 2020: UN Secretary General Antonio Guterres calls for a nationwide ceasefire in Yemen.
- April 2020: Foreign Office minister James Cleverly calls for the release of Mr Symons as Ramadan approaches, and highlights the risks from coronavirus in Yemen's unhygienic jails.
- September 2020: The internationally recognised Yemeni government and the Houthi rebels agree in Switzerland to exchange more than 1,000 prisoners in the largest swap of its kind in the conflict as ceasefire talks continue.
- October 2020: The exchange starts with the release of two US citizens. Planes take prisoners between the Houthi-controlled capital Sanaa and Aden city, the temporary capital of the Yemeni government.
- March 2021: UK Foreign Secretary Dominic Raab writes to Houthi leader Abdulmalik Al Houthi, urging him to release Mr Symons as the fourth anniversary of his arrest nears.