Three months into oil’s rebound from a six-year low, the global tanker market is sending clues that further upside momentum in crude prices may be limited.

A sudden surge in demand for supertankers drove benchmark charter rates nearly 60% higher at the end of May. Last year’s crude sell-off pushed the cost of oil for prompt delivery to well below future prices, making it profitable to purchase oil and store it to sale at a later time.

Off the coast of West Africa and in the waters of the North Sea, supertankers holding million barrels of oil have essentially become accidental storage as their owners seek customers.

The vessels are competing with new loadings, as well as time-chartered cargoes that major trading houses such as Trafigura and Unipec booked to store crude months ago and are now selling.

The release of floating oil volumes will introduce additional supply into an already saturated market and exacerbate the number of unsold Mediterranean, North Sea and West African cargoes in the Atlantic Basin. At the time of this writing, nearly 55 million barrels were being stored on tankers—more than double the amount in January, Thomson Reuters data show.

On March 16, U.S. benchmark West Texas Intermediate (WTI) crude fell 2% to near $44 per barrel (bbl)—a six-year low—and Brent crude, the international marker, erased about 2%, settling around $53 per bbl.

Crude futures on June 29 hit three-week lows as Greece shut its banks and imposed capital controls, causing widespread risk aversion, while Iran looked likely to extend nuclear negotiations with the West to export more of its oil into an oversupplied market. Brent crude closed down $1.25 to $62.01 per bbl a barrel, its weakest finish since June 5. U.S. crude finished at $58.33 per bbl, down $1.30—its lowest settlement since June 8.

Until Monday, Brent had been trapped in a $62 per bbl to $65 per bbl range, and WTI had traded within a $59 per bbl to $61 per bbl range.

Industry observers say the physical market for crude oil will remain under pressure as traders unwind profitable storage plays after several months that that saw them holding millions of barrels on tankers at sea.

Tanker Rates Explode

In another sign that oil is set to fall, the Baltic Dirty Tanker Index, which tracks the rates to hire oil tankers plying 16 routes, shot up 25% in June to its highest level since January 2014, the Wall Street Journal reported June 25.

“Tankers are making serious money, in what’s turning into the best tanker market since 2008,” Wells Fargo analyst Michael Webber was quoted as saying.

In fact, earnings for the gravity-defying tanker market remain strong ahead of the so-called traditionally week third quarter. Current rates for large crude tankers are between $40,000 and $50,000 a day. Rates for products carriers top $30,000 a day for LR1 (long-range) and LR2 fixtures to the Arabian Gulf/Far East, and MR (medium-range) tankers cost at least $20,000 a day.

Still, some analysts fear the robust market—spurred by what they believe is fleeting global demand—could be bearish for oil prices in coming months.

“There’s a lot of crude oil that’s trying to find a home, so that’s good for the tanker market, [but] that limits the potential for a crude-oil rally and puts pressure on the price,” Olivier Jakob, managing director of Switzerland-based Petromatrix, told the Journal.

Amrita Sen, co-founder and chief oil analyst at U.K.-based energy market consultancy Energy Aspects said the absolutely dire West African crude balances have finally weighed on Brent time spreads, with the contango out to January 2016 widening out, amid extensive liquidation.

“The disconnect between the markets is at unprecedented levels, and something has to give,” Sen observed in a June 23 research note.

Shipping analysts earlier this year credited higher oil demand for giving the crude tanker market a “stellar” start to 2015, with the expectation that 2016 will also be better than expected.

Homeless and Adrift

But there aren’t enough buyers for all the crude oil out there, according to Morgan Stanley’s Adam Longson.

“It’s peak season for oil buying, yet there are still a bunch of tankers full of oil sitting in the Atlantic Basin waiting to be sold,” Longson wrote in a June 22 client note. And when it comes to the future trajectory of oil prices, that is “a worrying sign for the fall,” he said.

“If there are this many challenged cargoes in this strong demand environment, we worry about the outlook for physical oil this fall when crude runs and gasoline demand fall seasonally,” Longson observed. “When combined with risk of new supply from Libya and Iran, a more range bound (if not lower), yet volatile, oil price environment seems increasingly likely in the second half of the year.”

Although some oil cargoes are being bought after months of floating in storage—namely Nigerian and North Sea crudes—the sheer volume of oil in the market has driven the prices of different crude oil grades to the lowest level in years, Longson observed.

As expected, OPEC maintained its oil output target at 30 million barrels per day at its latest summit in Vienna. At the same time, U.S. shale production has continued higher, though the Energy Information Administration now predicts output to slow beginning this month and continuing into early next year.

According to a recent Commerzbank note: “OPEC is currently producing 1 million barrels of crude oil per day more than will be needed in the second half of the year … It will therefore only be possible to reduce the oversupply if non-OPEC supply declines, though there is no indication for this to happen at the moment.”

So in Morgan Stanley and other industry observers’ view, even the strongest demand is not balancing the market.