STORY: Shipping firms say it could take weeks for movements to get back to normal in the Strait of Hormuz, despite the U.S.-Iran peace deal.
In a social media post, Donald Trump said tankers were already moving oil out of the area.
The U.S. president called the Strait "totally safe, secure and pristine".
:: MarineTraffic
But vessel tracking data showed no significant movement on Monday.
Just one liquefied natural gas carrier was seen making the transit.
However, that could understate the real numbers, with the data only capturing ships transmitting their position.
In recent weeks, some vessels have been switching off their transponders, or sailing "dark", and making the transit with the help of the U.S. navy.
For now, shipping firms welcome the deal, but also express caution.
Companies in Asia and Europe told Reuters they would only start moving once safety is assured, and the U.S. and Iran have a framework for control of the waterway.
They want to see more detail on moves including the clearance of any mines.
Some industry experts also noted that the Strait has been declared open before, only for little to change.
German shipping firm Hapag-Lloyd said it hoped to resume transits this week.
Danish giant Maersk said it was too early to assess the impact of the deal, and it was making no changes to Middle East operations for now.
Countries around the world will be watching, as around a fifth of the world's oil supplies normally moves through the Strait.
Industry experts say as many at 215 tankers are still stuck in the area.
