Should I or should I not?
Sigh, okay, here we go.
Muslims who don't touch women is not rooted in respect, it's rooted in women being perceived as impure because they bleed and because they are the property of another man and you don't touch the property of another man. There is no good faith behind it. It's very different from Asians who bow instead of shaking hands irrespective of sex. Asians will also not decline if you offer your hand as a Westerner. Because they know that's your culture, put their own behind for a moment out of respect and don't force their own view onto others at all costs at all moments.
Also, to not shake hands of women is common in Saudi Arabia, Iran, Afghanistan, Sudan and Yemen. The pillars of "respectul" countries and human rights. It's not a general Muslim rule in all Muslim countries. People in Turkey generally kiss on the cheeks to greet. Women included. If I asked people about this practice in Turkey they'd tell me they want nothing to do with backwards Arab culture.
Black DEI woman knowing her place and apologizing to Muslim man for daring to offer a shake hand in Munich is a weirdly accurate representation of current Western Europe though, I must admit.
Regarding the topic of lascivious western whores and good, protected muslim girls who can't be touched by a personal trainer, my own experience in Europe in muslim / migrant districts is that hymen reconstruction surgery is at an all time high. Me personally, when I was younger, had my dick inside quite a lot of muslim girls whose fathers kept calling to check where they were and they pretended to study at their girlfriends houses while cleaning up my semen from their mouths. They are most likely married to some Muslim men now who believe they married a virgin. Taking it in the ass "to stay virgin" was also quite common. "The forbidden" turning them into whores even more than Western girls should surprise no one, it's expected psychologically. The forbidden always has the highest temptation. A lot of famous American pornstars come from strict religious Texan "girls schools only / homeschooling" type households. No one is holier than thou.
Muslim women who are treated as property like "you don't touch my Lambo, bro" defending this is also psychologically expected. First, women are far more herd animals than men. Evolutionary, women stepping out of their group was a death sentence. Men had higher survival chances on their own. Men are more likely to go against the grain or do their own thing as loners. Women are far more social creatures because they had to evolutionary. That's the main reason why men invented pretty much everything substantial. It's not possible when everyone just follows the established group think. Earth is not the center of universe, light is neither particle nor wave only, time and space is relative etc. Insights completely against the norm at the respective times. That's why women in conservative times like the 50s tend to be even more conservative than men ("oh, look how the neighboor's wife dresses today, what a whore") while in woke times like nowadays they are the biggest proponents of rainbow flags, endless migration and whatever else the mainstream upholds today. In short: Herd animals. So in Muslim countries they'll defend the culture surrounding them to death despite being victims themselves.
On top of that you have Stockholm syndrome going on. Deep down you would like to be free like the Western girls but your father, brother, uncle and everyone in your group will never allow it to happen, in worst cases they will kill you to restore the family's honour, so what's your choice really? You have to defend this culture to protect your own psyche. No one can live happy being surpressed unless you manage to not be surpressed and flee the country as well as your family or pretend that you are not surpressed and start defending your surpressors as protectors and care takers.
I know this will offend my Muslim brethren here deeply but none of that is meant as an insult. I'm merely telling my observations. Do you know why the 10 million you hate so much dominate the 1.5 billion of your lot? Because the 10 million invented quantum physics, nuclear weapons and technology 100 years ago, meanwhile the 1.5 billion to this day keep knocking their heads against the floor 5 times a day and take pride in not shaking hands, hoping for some deity to save them in the afterlife for doing so.