Two supertankers, each capable of hauling about 2 million barrels of crude, headed through the Hormuz shipping strait at the mouth of the Persian gulf, having performed U-turns in the past 24 hours.

On Sunday, the Coswisdom Lake and South Loyalty entered the waterway and abruptly changed course, according to vessel tracking data compiled by Bloomberg. After a pause, both then headed back through Hormuz. One was almost through and the other entering it, signals from each showed on Monday.

The two vessels’ movements into the world’s most important oil-producing region come as shipbrokers and others in the oil tanker market report a cautious willingness to enter the stretch of water that’s pivotal to the global oil trade. Separately, Greece’s shipping ministry warned on Sunday that the country’s owners should think twice about Hormuz transit and instead head to a safe port.

While neither tanker is Greek, the communication from Athens underscored an initial wariness about going through an area that handles about a fifth of the world’s oil. Greece is the owner of the world’s largest tanker fleet by transportation capacity.

Coswisdom Lake is managed by Cosco Shipping Energy and South Loyalty by Sinokor Merchant Marine, according to industry databases. Both are very large crude carriers, or VLCCs. Neither company immediately responded to emailed requests for comment.

There remain examples of wariness. Two large large Japanese shipping companies said they will cut exposure to the strait, an unavoidable searoute any vessel entering the Persian Gulf.