First, oil inventory data. This is from the privately surveyed oil stock data ahead of official government data tomorrow morning out of the US.Via oilprice.com:Expectations I had seen centred on:Headline crude -3.6mn barrelsDistillates -0.6 mn bblsGasoline +3.5 mnAs for Tuesday, oil higher. Brent settled at $96.00 and WTI at $93.76, one-week highs, as conflicting Iran-US signals over a proposed ceasefire MoU kept oil volatile; the Strait of Hormuz remained closed.SummaryBrent rose 1.1% to $96.00 and WTI gained 1.7% to $93.76, the highest closes for both benchmarks since May 26Iran's semi-official Fars agency reported that the exchange of messages between Tehran and Washington toward an initial MoU had halted for at least several days; the trigger for the session's initial oil price strengthTrump and Secretary of State Rubio pushed back, with Rubio stating the war in Iran is over and Trump saying reports of a communications halt were false and that talks had been continuousAmwaj Media, citing a senior Iranian political source, said communications via intermediaries had not stopped and that movement across multiple tracks suggested a deal could be closeOn the Israel-Lebanon front, sources cited by the Israeli Broadcasting Authority said negotiations were progressing, while Hezbollah rejected any partial ceasefire; the US is pressing for a comprehensive deal by end of Wednesday's negotiating roundIraq announced plans to raise pipeline crude exports from 220k BPD to 770k BPD across two phases over two and a half months, and to lift truck exports to neighbouring countries to 420k BPD in three phasesThe IEA warned global inventories could reach critical lows before peak summer demand if draw rates persist; analysts forecast a fourth consecutive US crude stock draw of around 4 million barrels for the week ended May 29Oil prices ground steadily higher through Tuesday's US session, settling around 1% firmer at one-week highs as traders tried to navigate a stream of contradictory headlines on the Iran-US conflict and its implications for the blocked Strait of Hormuz.Brent crude closed $1.02 higher at $96.00 a barrel, while West Texas Intermediate added $1.60 to settle at $93.76. Both benchmarks posted their best closes since May 26 in what remained choppy, headline-driven trade throughout the day.The session's initial upward push was sparked by a report from Iran's semi-official Fars agency, which cited informed sources saying the exchange of messages between Tehran and Washington, aimed at brokering an initial memorandum of understanding, had been halted for at least several days. That reading of proceedings was quickly challenged from multiple directions. Secretary of State Marco Rubio told US lawmakers that talks with Iran were active and went so far as to say the war in Iran was over, while Amwaj Media, citing a senior Iranian political source, said communications via intermediaries had in fact continued and that progress across several parallel tracks suggested the two sides could be nearing something concrete. President Trump weighed in to call reports of a communications halt false, insisting conversations had been going on continuously while adding that Iran needed to make a deal regardless.The Iran-Lebanon dimension remained a sticking point. Tehran has conditioned progress on a halt to Israeli operations against Hezbollah, and Israel continued strikes on southern Lebanon on Tuesday despite Trump having asked Prime Minister Netanyahu to hold back from attacking Beirut. The Israeli Broadcasting Authority cited sources saying Israel-Lebanon negotiations were moving forward, though Hezbollah flatly rejected any partial ceasefire arrangement. The US is pushing for a comprehensive deal to emerge from Wednesday's negotiating round.Energy advisory firm Ritterbusch and Associates captured the market mood well, noting that conflicting commentary across the White House, Tehran and Jerusalem was keeping crude gyrating, and that a meaningful reopening of Hormuz appeared no closer than it had two months ago. The strait has been largely shut to non-Iranian shipping since the conflict began more than three months ago, cutting off around a fifth of global oil and LNG flows and driving prices more than 50% higher.On the supply side, Iraq offered a modestly constructive offset, announcing plans to lift pipeline crude exports from 220,000 BPD to 770,000 BPD across two phases over the next two and a half months, alongside a phased increase in truck exports to neighbouring countries to 420,000 BPD. The volumes are meaningful but unlikely to compensate materially for Hormuz disruption at scale.The IEA added to the bullish inventory backdrop, with the head of its oil markets division warning that global stockpiles could hit critical levels before the summer demand peak if current draw rates hold. US analysts were forecasting a roughly four-million-barrel crude draw for the week ended May 29, which if confirmed by the API and EIA would represent the first run of six consecutive weekly draws since January 2025. This article was written by Eamonn Sheridan at investinglive.com.