While General Mills' Yoplait brand has made some progress in Greek-style yogurt, it's still an "eventual loser" in the Greek market, says a report this week from Bernstein Research.

However, the Greek yogurt juggernaut is slowing down some, which should help Yoplait's conventional yogurt business, the report said.

Greek-style yogurt makes up 49 percent of the U.S. yogurt business, up from only 1 percent in 2007. Yogurt upstart Chobani led Greek's meteoric ascent and has 35 percent of the Greek market.

The U.S. yogurt business's traditional top two, Yoplait and Dannon, were late to the Greek party.

But Dannon, which is owned by France-based Groupe Danone, has made up more ground than Yoplait. Dannon has captured 31 percent of the Greek market, taking share from Chobani over the last year, according to Bernstein.

Yoplait had 9 percent of the Greek market for the four weeks ending April 12, according to the Bernstein report.

Golden Valley-based General Mills, which trotted out a particularly successful 100-calorie Greek product, gained share in 2012 and 2013. But Yoplait's market share progress in Greek has stalled in recent months, the Bernstein report said.

As Greek style yogurt has grown in popularity, it's eaten into the sales of conventional yogurt, a bad situation for Yoplait.

General Mills overall share of the U.S. yogurt market – both Greek and traditional included – has fallen from 42 percent in 2008 to an estimated 24 percent in 2014, according to Bernstein.

But as sales growth in Greek-style yogurt growth has slowed some, the decline in conventional yogurt sales has also tapered. "The negative growth is getting less bad," the Bernstein report said.