Broadway musical “Priscilla Queen of the Desert” will shutter in June after a run of about 16 months.The Gotham incarnation of the tuner — which bowed in Sydney, played the West End and recently opened in Rio de Janeiro — posted robust sales in its initial weeks on the boards, but then levelled off into generally solid but often unspectacular numbers. This spring B.O. has continued to taper off somewhat, landing south of the $600,000 mark in recent weeks. Last week the production logged $535,357 from attendance at 53% of capacity. Show began previews Feb. 28, 2011, and opened March 20. Production has earned about $45 million in its time on the boards. Musical about a trio of Aussie drag queens on a road trip draws its score from familiar disco-era hits. Will Swenson, Tony Sheldon and Nick Adams star in the New York staging, which also snagged a Tony for costume design. Bette Midler, James L. Nederlander, Garry McQuinn and Liz Koops are among the producers of “Priscilla.” Show goes out on a national tour early next year, launching in Minneapolis on Jan. 8. Tuner shutters June 24 after a run of 23 previews and 526 regular perfs.