Despite Yemen’s ceasefire holding, no progress has been made on opening roads into Taez, which the UN on Monday said was cutting off aid to millions of people in need.
The main players in the country's civil war this month agreed to extend a two-month ceasefire, which was first introduced on April 2, for a second time.
Part of the truce renewal agreement was lifting the Houthi siege on Taez province and the resumption of flights from the rebel-held airport in the capital Sanaa.
“It is regrettable that, despite these efforts, there has not been more progress achieved on road openings to date,” Hans Grundberg, the UN envoy to Yemen, told the Security Council.
Mr Grundberg urged both sides, for “the sake of the people of Taez, the wider population and the economy, to agree on opening roads as soon as possible”.
The opening of routes to the besieged city is “mainly a humanitarian issue and the truce provides a conducive environment for the parties to swiftly deliver on this issue”, he said.
The people of Taez and across Yemen “deserve for the truce to deliver for them in all its aspects”, he said.
Yemen’s internationally recognised government accused the Iran-backed rebels of wilfully failing to open roads to Taez in a breach of a UN-brokered truce.
More than four million Yemenis have been affected by the Houthi blockade, the country’s Foreign Minister Ahmed bin Mubarak said last week.
The routes to Taez are punctuated by roadblocks and detours and the cost of transport can quadruple along the way, complicating the delivery of aid and depriving many of access to basic services.
The city, which was once an important cultural, academic and historical centre, is split by a 16-kilometre-long front line.
About 80 per cent of its population lives in the government-held part of Taez but the rebels control the higher ground where the city's water wells are located.
Mr Grundberg said his team has reported a significant decline in civilian casualties, and the lowest count since the start of the truce in April was registered in the first week of this month.
“Four and a half months in, the truce continues to broadly hold in military terms,” he said. "No major military operations or changes to front lines have occurred and there have been neither confirmed air strikes inside Yemen nor cross-border attacks emanating from Yemen."
However, the UN diplomat said the number of child casualties continued to increase.
“One particularly horrific incident occurred in Taez on July 23, when a mortar was fired into the residential Zaid Al Mushki district, killing one child and injuring 10 others,” Mr Grundberg said.
“I condemn all such acts of violence and civilians must be protected at all costs.”
The Houthi rebels seized control of Sanaa in 2014, prompting a Saudi Arabian-led military intervention to support the internationally recognised government the following year.
Hundreds of thousands of people have died in the civil war to date.