Erica Howton : Many thanks for making the effort on quoting (Riley-Smith, Jonathan, The First Crusaders, 1095-1131, Cambridge University Press, Cambridge, 1997. Page 172.) Riley.
This will help you and Sharon Doubell if you read the full book to understand the arguments as multiple theories of discussions by Riley, which is always good for brainstorming. As you may have noted, this is a publication in 1997 which was the first to challenge the facts that Bailan le Francois was not the son of Hugh I of Le Puiset (died 23 December 1094), son of Everard I of Breteuil and his wife Humberge. Riley made a bold statement in 1997 against the re-produced publication of 1915 from 13th century manuscripts (that he had not come across then in 1997) and produced no evidence to prove the theory that he called myth as pointed out by you from his statement in the publication of year 1997. Later Riley-Smith speculates that Barisan may have been the illegitimate brother or brother-in-law of Gilduin in the same publication [ref: Riley-Smith, Jonathan. The First Crusaders. pp. 172–173] but there is little evidence to support these theories as well in year 1997.
Further, regarding Gilduin of Le Puiset: --> Gilduin of Le Puiset (d. between 1130 and 1135) and Riley claiming never existed. Well Guildin existed and he is documented as was the son of Hugh I of Le Puiset and Alice of Montlhéry, daughter of Guy I of Montlhéry. Monk at St. Martin-des-Champs, prior at Cluny Abbey, prior at Lurey-le-Bourg, abbot of St. Mary of the Valley of Jehosaphat. [Ref: Riley-Smith, Jonathan (1997). The First Crusaders. pp. 169–170]. Further, Gilduin became abbot of St. Mary, and was in this position when his cousin Baldwin II confirmed the privileges of the abbey, the chief Marian shrine in Jerusalem, on 31 January 1120. He was also documented in Families d'outre mer: as Viscount de Chartres (duration not specified, could be for a week or less - no idea). There is also a document of the year 1028 that states the above and this is referenced in the 1915 re-produced publication.
I suggest to reorganise the Ibelin tree to its original form for accuracy as users are going to have wrong information and connections with the mess created yesterday by multiple MP (main profiles) of same person and represented in two generations without evidence to substantiate those profiles.
I hope the above helps a little bit. :)