In the pedigree of the Stanhope Mare the GSB states "out of Mr Burdett's Young Child Mare by Sir J Harpur's Barb" [GSB 1:200]. However, Young Child Mare was sired by Burdett's Arabian, so if Harpur's Barb and Burdett's Arabian are the same horse the conflict is resolved. The family of Robert Burdett, the owner of Burdett's Arabian, lived near Calke Abbey in Derbyshire, the seat of the Harpurs, so this is very possible

Harpurs Barb sires a son Rattle, which sires an unnamed mare which is dam of the highly significant broodmare Silvertail, dam of 1) the outstanding racehorse and successful sire [Warren's] Careless, winner of 10 King's Plates between 1756 and 1758, and of 2) Fearnought, winner of 3 King's Plates in 1761 and an immensely successful stallion in America, the most important of the colonial era