Oil prices edged higher on Tuesday, though gains were capped as traders ‌looked beyond easing geopolitical tensions in the Middle East and turned their attention to supply increases and demand prospects.

Brent crude futures gained 38 cents, or 0.5%, to $72.37 a barrel, while US West Texas Intermediate crude rose 30 cents, or 0.4%, to $68.85 a barrel as of 0350 GMT, after settling down at ​around pre-Iran war levels on Monday.

"The steps towards recovery in supply have eased the immediate risk premium, but the market ​remains wary of putting too much faith in the stability of the current truce given the on ⁠again-off again nature of US-Iran relations," said Tim Waterer, chief market analyst at KCM Trade.

"We will be watching for early signs of ​demand response, particularly from China. The market has priced in a lot of the positive supply news, so the next leg in oil ​prices will depend on whether physical reality matches the optimistic headlines."

President Donald Trump said on Monday the US would either reach a deal with Iran or "finish the job," renewing his threat of military action as Tehran projects defiance following the funeral of former supreme leader Ayatollah Ali Khamenei.

Investors have been keeping a close eye on ​talks between the US and Iran over the fate of shipping through the Strait of Hormuz while tracking the recovery in Gulf ​oil exports.

On Monday night, Iran's Revolutionary Guards fired at least two missiles at commercial ships transiting the Strait of Hormuz, Axios reported, citing two US officials. ‌The ⁠commercial ships suffered significant damage but had no casualties, the report said.

Read: OGRA calls meeting with oil industry to break stalemate

On Tuesday, Japanese-owned supertankers carrying Saudi Arabian crude were heading to the Strait of Hormuz to exit the Gulf, shipping data showed, joining a fleet of previously stranded vessels that left a day earlier.

Despite the recent surge in strait activity, oil flow recovery is proving slower than expected, ANZ analysts said in a note.

"The initial rebound in tanker ​transits through the Strait of ​Hormuz has stalled, with vessel ⁠crossings remaining in single digits and no sustained recovery evident," they said.

"While the interim US-Iran agreement has reduced immediate geopolitical risks, shipping operators remain cautious, limiting the speed at which crude exports can return ​to normal levels".

Meanwhile, the United Arab Emirates raised crude output above 3.8 million barrels per day in ​June, its highest ⁠since April 2020 and above pre-Iran war levels, after leaving OPEC+ production quotas in May, according to Reuters estimates.

The Organisation of the Petroleum Exporting Countries and its allies, including Russia, agreed on Sunday to further increase output targets by 188,000 bpd from August, on top of similar increases ⁠for June ​and July.

Saudi Arabia cut the August official selling price (OSP) for its flagship Arab Light ​crude to Asia to $1.50 a barrel below the Oman/Dubai average, an $11 cut from the previous month and the biggest drop in more than two decades, according to a ​Saudi Aramco pricing statement on Monday.