Where is this happening as I have not seen it.
The liberalizing isn't near the levels of Christianity and I doubt it ever will be.
Are there any videos of these liberalist Muslims and the modern types of mosques with women running things that you speak of? And Muslims speaking in acceptance of even homosexuality and marriage of Muslim women to non-muslims?
The real litmus test is how many are able to renounce Islam altogether if they want to without threats or social exclusion. That is freedom of religion.
http://formermuslimsunited.org/apostasy-from-islam/
"If you covert you die...the penalty for leaving Islam is death in all schools of Sharia, both Sunni and Shiite...Islam promises heavenly rewards to individual Muslims who take the law into their own hands. Sharia states that the killers of apostates and adulterers are not to be punished as murderers"....
"Muslims are not just distrustful of the West, but they are distrustful of one another. People are often more afraid of their neighbors and family members than the police. Thus we see husbands or father pressured to apply Sharia by killing an adulterous wife or daughter, or a perfect stranger participate in killing of an apostate in the public square....The power of Islam comes from turning Muslim against Muslim with a reward in heaven."
In Europe in any case it's definately not like that in the main part. They generally will try to segment themselves into ghettos and follow their own laws where they can. In some cases this goes to the extreme including committing 'honor' killings on members of their own families or communities or others for what they perceive as misbehaving or not complying. Again it's not a majority by any means at least in Western countries, but a signifigant percentage still. The France teacher killing is another example.
And like it or not it's had some effect.
https://www.rt.com/op-ed/511817-teachers-lessons-offend-muslims/