Context: The first major battle of the war took place at Casal Imbert in May 1232. Filangieri defeated the Ibelins. In June, however, he was so soundly defeated by an inferior force at the Battle of Agridi in Cyprus that his support on the island dwindled to zero within a year. In 1241 the barons offered the bailliage of Acre to Simon de Montfort, the Earl of Leicester, a cousin of Philip of Montfort, and a relative through marriage to both the Hohenstaufen and the Plantagenets. He never assumed it. In 1242 or 1243 Conrad declared his own majority and on 5 June the absentee monarch's regency was granted by the Haut Court to Alice, widow of Hugh I of Cyprus and daughter of Isabella I of Jerusalem. Alice promptly began ruling as if queen, ignoring Conrad, who was in Italy, and ordering Filangieri arrested. After a long siege, Tyre fell on 12 June. The Ibelins seized its citadel on 7 or 10 July, with the help of Alice, whose forces arrived on 15 June. Only the Ibelins could claim to be the winners of the war.

Question: Who lost the first major battle of the war?

Answer:
Ibelins