Does necessity permit shaking hands with a non-mahram woman to avert temptation or avoid societal accusations in countries where this custom prevails?

Among the widespread reprehensible practices is a man shaking hands with a non-mahram woman, and this is forbidden (haram). If a woman extends her hand to shake yours, withhold your hand and gently apologize, explaining that this is not permissible in your religion. Any perceived harm you fear can be averted by apologizing and clarifying. It is not permissible to abandon the commands of the Sharia based on assumptions and probabilities.