Although the Curonians had attacked Riga in 1201 and 1210, Albert of Buxhoeveden, considering Courland a tributary of Valdemar II of Denmark, had been reluctant to conduct a large scale campaign against them. After Albert's death in 1229, the crusaders secured the peaceful submission of Vanemane  by treaty in 1230. In the same year the papal vice-legat Baldouin of Alnea annulled this agreement and concluded an agreement with the ruler of Bandava in the central Courland Lamekins , delivering his kingdom into the hands of the papacy. Baldouin became the popes's delegate in Courland and bishop of Semigallia; however, the Germans complained about him to the Roman Curia, and in 1234 Pope Gregory IX removed Baldouin as his delegate. After their decisive defeat in the Battle of Saule by the Samogitians and Semigallians, the remnants of the Swordbrothers were reorganized in 1237 as a subdivision of the Teutonic Order, and became known as the Livonian Order. In 1242, under the leadership of the master of the Livonian Order Andrew of Groningen, the crusaders began the military conquest of Courland. They defeated the Curonians as far south as Embūte, near the contemporary border with Lithuania, and founded their main fortress at Kuldīga. In 1245 Pope Innocent IV allotted two thirds of conquered Courland to the Livonian Order, and one third to the Bishopric of Courland. At the Battle of Durbe in 1260 a force of Samogitians and Curonians overpowered the united forces of the Livonian and Teutonic Orders; over the following years, however, the Crusaders gradually subjugated the Curonians, and in 1267 concluded the peace treaty stipulating the obligations and the rights of their defeated rivals. The unconquered southern parts of their territories  were united under the rule of the Grand Duchy of Lithuania.

Based on the above article, answer a question. Who ruled the unconquered southern parts of their territories?
Lithuania