Apple’s newly announced iPhone SE could be less popular than the company hopes, according to a new study.

Apple AAPL will sell between 4 million and 6 million iPhone SE units in the U.S. in its first year of availability, Consumer Intelligence Research Partners (CIRP) wrote in a study released this week. CIRP added that it’s unlikely for current owners of Apple’s larger iPhones to “trade down” to the iPhone SE.

“The analysis indicates the limited opportunity for Apple to sell the smaller iPhone SE in the U.S.,” Mike Levin, partner and co-founder of CIRP, said in a statement. “Very few of the 81 million consumers that already own a larger iPhone 6 or 6 Plus, or 6s or 6s Plus, are likely to trade down to the smaller iPhone SE.”

The other issue, says Josh Lowitz, Levin’s partner, is that CIRP’s survey of 500 Apple customers shows that the vast majority of current iPhone owners may be more interested in going up in screen size, rather than sticking with a 4-inch display.

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“A significant number of the 29 million consumers that currently own an older iPhone will also want to trade up to the larger phones,” Lowitz said in a statement. “Our data suggests in the past year, 85-90% of consumers that replaced an older 4-inch screen iPhone upgraded to the new, larger iPhone models—the 6, 6 Plus, 6s, or 6s Plus. So, we expect perhaps 3-4 million, or 10-15% of the 29 million US 4-inch iPhone base, to purchase the new iPhone SE.”

Apple announced its long-rumored iPhone SE at a special press event Monday. The device comes with a nearly identical design to that of the iPhone 5s, the smartphone the company launched in 2013 and had been selling until the SE came along. Despite the design similarities, the iPhone SE boasts a better processor and camera than the iPhone 5s.

Perhaps most important, the device is cheaper than the iPhone 5s. While Apple was previously selling its decidedly underpowered iPhone 5s for a starting price of $450, the iPhone SE starts at $399.

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It’s important to note, therefore, that CIRP’s data may be comparing two entirely different things. Customers may have been choosing the larger-screened iPhone 6 or iPhone 6s over the last couple of years due to iPhone 5s power issues compared with its alternatives. And with a more affordable price tag, it’s possible that Apple could attract more shoppers to the iPhone SE than it did to the iPhone 5s.

“It was a bit of a surprise that [Apple] dropped the price of the entry-level iPhone more than 10%, even as they copied over most of the cool new technology of the iPhone 6s,” Forrester Research vice president and principal analyst Frank Gillett told Fortune on Monday. “Getting below $400 is a new level that will draw interest from buyers who thought they could only afford Android.”

He added that the company’s new strategy on pricing could help it “reach a lot of people.”

Indeed, the U.S. isn’t the only market that Apple cares about. The company was quick to note at its “Let us loop you in” event Monday that while the iPhone SE will be available to a handful of countries this month, it plans to sell it in 110 countries by the end of May. It’s in those countries—especially emerging markets where price sensitivity is a concern—that the iPhone SE could be successful.

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“The smaller iPhone SE is targeting existing Apple customers who prefer the old 4-inch display size, as well as first-time iPhone users as a lower priced entry point into the Apple ecosystem,” Ben Schachter, an analyst at Macquarie Securities, wrote to investors Monday.

In the U.S., though, CIRP believes the possibility of first-time phone buyers and even people looking to switch from an Android device to the iPhone SE will be relatively slight. The company said that in the past year, across all iPhone models, just 17% of those who own an Apple smartphone previously owned an Android handset. What’s more, just 1% of all iPhone customers in the U.S. are first-time phone buyers, CIRP says.

“We also think that relatively few Android switchers and first-time phone buyers will opt to purchase the iPhone SE,” Levin said.

CIRP’s findings are just the latest to throw cold water on the iPhone SE’s chances of success. Several analysts told Reuters on Tuesday that based on their understanding of market demand in key markets such as China and India, the iPhone SE could be a flop.