Kirk,
I too have been working on attempting to answer a similar question and contacted a New England Historic Genealogical Society genealogist for assistance. To build upon what others have researched and posted:
Background: Earlier genealogical research was often accepted based on loose assumptions (See: https://www.wikitree.com/wiki/Space:Cochoit_Working_3). Subsequently, Constant Southworth’s potential royal descent has not been definitively proven. I hope that by combining additional research, analysis, and Y-DNA results through projects like the Family Tree DNA Southworth Surname project (https://www.familytreedna.com/groups/southworth-southard/about/background) the ancestry of Constant Southworth will be resolved.
Additional Research and Analysis Since 1992: Under Gary Boyd Roberts, The Royal Descents of 900 Immigrants to the American Colonies, Quebec, or the United States (Baltimore: Genealogical Publishing Co., 2018), 1:708-709, Mr. Roberts first acknowledges the uncertainty of Edward’s origins: “Edward Southworth, probably [emphasis added] the Leyden Pilgrim…” (p. 708). He then addresses Sue Allan’s “In Search of Separatist Edward Southworth of Leiden” 2017 online monograph https://shop.americanancestors.org/products/in-search-of-separatist-edward-southworth-of-leiden-his-genealogical-origins-uncovered?pass-through=true (2019 version) argument against Leyden as Edward’s correct origin as Edward’s father, Thomas Southworth of Samelsbury, Lancashire was a Protestant, while Thomas’s father, Sir John Southworth, and two of Thomas’ brothers were “sometime ardent Catholics variously imprisoned.” Alternatively, Ms. Allan presents a circumstantial argument that Edward Southworth of Leyden was born 12 April 1585 at Clarborough, Nottinghamshire, son of Richard Southworth of Clarborough and Immyn Aston; and this Richard was son of Richard Southworth of Clarborough and Welham and Emma Levesey of Keeton (in Nottinghamshire).
Resolution Strategy: Identify descendants of Sir John Southworth and Mary Ashton and compare DNA results against descendants of Thomas Southworth (including grandsons Constant and Thomas Southworth) (Minimum 67 markers, possibly upgrade to 111 markers or Big Y-DNA). Identify living male descendants of Richard Southworth and his wife Imogene Aston of Clarborough, Nottinghamshire. According to the Southworth genealogy, Richard had ten children. Of them were a Thomas born 28 July 1583 and Edward born 12 April 1585. Visitations of the County of Nottingham in 1569 and 1614: - https://babel.hathitrust.org/cgi/pt?id=yale.39002005263265&view=2up&seq=130
Richard Southworth who married Imogene Aston was the son of Richard Southworth of Clarborough who married Emma Levesey. Richard and Emma’s son Robert is named in the visitation books along with his children. Going through Richard’s line to find living male descendants may be a good way to test this line.
Additionally, I am looking for direct male lineal cousins who descend from Moses Southworth (1835-1887) - my last direct lineal male descendant - who are interested in taking a DNA test and sharing it with the Family Tree DNA Southworth-Southard Surname project. Please let me know if you or others reading this post are interested in collaborating.