Question: What cousin of Herman I, Duke of Swabia also had a chance to rule the Duchy of Franconia?

Context articles: Conrad I, Duke of Swabia
Herman I, Duke of Swabia
Frederick III, Duke of Upper Lorraine
Duchy of Croatia
Herman IV, Duke of Swabia
Duke of Swabia
Conrad I of Germany
Ludwig II of Bavaria
Franconian Circle
Lothair II, Holy Roman Emperor

Conrad I (also Konrad) (born 915/920 - died August 20, 997) was Duke of Swabia from 983 until 997.
 His appointment as duke marked the return of Conradine rule over Swabia for the first time since 948.
Herman I (died 10 December 949) was the first Conradine Duke of Swabia (from 926), the son of Gebhard, Duke of Lorraine, and a cousin of King Conrad I of Germany.
Frederick III ( 1020 – 1033) was the Count of Bar and Duke of Upper Lorraine from the death of his father, Frederick II, in 1026 or 1027 to his own death.
 His mother was Matilda of Swabia, daughter of Herman II, Duke of Swabia.
"Duchy of Croatia" (Croatian: "Kneževina Hrvatska" ; also "Duchy of the Croats", "Kneževina Hrvata"; "Dalmatian Croatia", "Dalmatinska Hrvatska"; "Littoral Croatia", "Primorska Hrvatska"; Greek: Χρωβατία, "Chrovatía"), was a medieval Croatian duchy that was established in the former Roman province of Dalmatia.
 Throughout its time it had several seats, namely Klis, Solin, Knin, Biaći and Nin, comprised the "littoral", or coastal part of today's Croatia and included a big part of the mountainous hinterland.
 The Duchy was in the center of competition between the Carolingian Empire and the Byzantine Empire for rule over the area.
 Rivalry with Venice emerged in the first decades of the 9th century and was to continue for the following centuries.
 Croatia also waged battles with the Bulgarian Empire, with whom the relations improved greatly afterwards, and the Arabs and sought to extend its control over important coastal cities under the rule of Byzantium.
 Croatia saw periods of vassalage of the Franks or Byzantines and de facto independence until 879 when Croatian Duke Branimir received recognition from Pope John VIII as an independent realm.
 The ruling dynasty of Croatia was the House of Trpimirović, with interruptions by the House of Domagojević (864–878 and 879–c. 892).
 The Duchy existed until around 925 when, during the rule of Duke Tomislav, Croatia became a kingdom.
Herman IV (c. 1015-July 1038) was the Duke of Swabia (1030–1038).
 He was the second son of Ernest I and Gisela of Swabia.
 He was one of the Babenberg dukes of Swabia.
The Dukes of Swabia were the rulers of the Duchy of Swabia during the Middle Ages.
 Swabia was one of the five stem duchies of the medieval German kingdom, and its dukes were thus among the most powerful magnates of Germany.
 The most notable family to rule Swabia was the Hohenstaufen family, who held it, with a brief interruption, from 1079 until 1268.
 For much of this period, the Hohenstaufen were also Holy Roman Emperors. With the death of Conradin, the last Hohenstaufen duke, the duchy itself disintegrated, although King Rudolf I attempted to revive it for his Habsburg family in the late-13th century.
Conrad I (German: "Konrad" ; c. 881 – December 23, 918), called the Younger, was the first non-Carolingian king of East Francia from 911 to 918.
 He was the first elected king of East Francia and also the first one to be anointed.
 He was chosen as the king by the rulers of the East Frankish stem duchies after the death of young king Louis the Child.
 Prior to this election he had ruled the Duchy of Franconia from 906.
Ludwig II (German: "Ludwig Otto Friedrich Wilhelm" ; English: Louis Otto Frederick William ; 25 August 1845 – 13 June 1886) was King of Bavaria from 1864 until his death in 1886.
 He is sometimes called the Swan King, Mad King Ludwig or der Märchenkönig (the 'Fairy Tale King').
 He also held the titles of Count Palatine of the Rhine, Duke of Bavaria, Duke of Franconia, and Duke in Swabia.
The Franconian Circle (German: "Fränkischer Reichskreis" ) was an Imperial Circle established in 1500 in the centre of the Holy Roman Empire.
 It comprised the eastern part of the former Franconian stem duchy — roughly corresponding with the present-day Bavarian "Regierungsbezirke" of Upper, Middle and Lower Franconia — while western Rhenish Franconia belonged to the Upper Rhenish Circle.
 The title of a "Duke of Franconia" was claimed by the Würzburg bishops.
Lothair II or Lothair III (before 9 June 1075 – 4 December 1137), known as Lothair of Supplinburg, was Holy Roman Emperor from 1133 until his death.
 He was appointed Duke of Saxony in 1106 and elected King of Germany in 1125 before being crowned emperor in Rome.
 The son of the Saxon count Gebhard of Supplinburg, his reign was troubled by the constant intriguing of the Hohenstaufens, Duke Frederick II of Swabia and Duke Conrad of Franconia.
 He died while returning from a successful campaign against the Norman Kingdom of Sicily.