According to a licensing board page within the City of Boston website, Southern Proper is planning to open on Harrison Avenue, with the liquor license being transferred from the now-closed Williams Tavern in South Boston (pending approval from the city). The manager listed is Jason Cheek who comes from such restaurants as The Maiden, Sam's at Louis, Merrill & Co., Coppa, KO Prime, and Toro, and based on the name of the proposed spot--and the fact that Cheek has roots in the South--it appears that the new eatery could be focusing at least in part on Southern-influenced dishes.
Currently, the only information known about Southern Proper is that it would be located within The Girard (a new mixed-use building which includes luxury apartments), with the spot having a dining room and a bar on the ground floor along with a seasonal patio.
The address for Southern Proper is 600 Harrison Avenue, Boston, MA, 02118.
[October 1 update: Eater Boston states that Southern Proper will indeed be a Southern-style restaurant, but it will not be "kitschy," nor will it be a barbecue joint, though it will probably have some smoked foods and BBQ.]
[February 27, 2018 update: A Facebook post from the place indicates that Southern Proper opens to the public on Thursday, March 1.]
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[A related post from our sister site (Boston's Hidden Restaurants): List of Restaurant Closings and Openings in the Boston Area]