Why didn't South Tyrol give it to Austria?

After the defeat of World War I, the Austro-Hungarian Empire split. According to the postwar agreement, North Tyrol and East Tyrol still belong to Austria, but South Tyrol and Welsch Tyrol were forcibly assigned to Italy against the wishes of local residents, becoming the Trentino-South Tyrol district in northern Italy, which is why the same area belongs to two countries. Today, South Tyrol has become an unforgettable pain in Austria's heart. Many Tyroleans on both sides of the border still regard South Tyrol as a part of Greater Tyrol, and nostalgic Austrians do not forget to summarize the information of star-rated hotels in South Tyrol in the local resort hotel column of the newspaper, and even broadcast the weather situation in South Tyrol in the daily weather forecast. When it comes to Italy, the first thing people may think of is that it dragged the Germans back many times in World War II, but they don't know that it was Italy that seriously cheated another good friend, the Austro-Hungarian Empire, in World War I, which led to the loss of the Austrians in South Tyrol.