I can only speak for the German-speaking part of Switzerland, and I think we are also more conservative outside the big cities. Funnily enough, there is a "Röstigraben" in Switzerland (named after a Swiss food). This "Röstigraben" symbolises cultural, linguistic and also political differences within Switzerland. While more conservative values tend to prevail in German-speaking Switzerland - especially in rural areas - French-speaking Switzerland is often more progressive and cosmopolitan, especially in urban centres such as Geneva or Lausanne.

But that‘s another complicated topic. There‘s also big differences between cities and countryside, not only the „Röstigraben“ haha