Does divorce occur if a man tells his wife, "If you shake hands with any non-mahram, then every handshake counts as a divorce," and then the wife shakes hands out of duress to avoid embarrassing her relatives?

The majority of scholars hold that a conditional divorce takes effect if the condition is met. So, if you made your wife's divorce conditional upon her shaking hands with a non-mahram (definition) man, and this occurred three times, then a major irrevocable divorce has taken place, and she becomes unlawful for you until she marries another man. The occurrence of the second and third divorces is contingent on the wife still being under your marital authority. Whereas, Shaykh al-Islam Ibn Taymiyyah and those who agreed with him hold that the divorce does not take effect, and you are obligated to offer an expiation for a broken oath. It is important to emphasize the prohibition of a woman shaking hands with a non-mahram man.