Liberals have a hope that the same approach used to water down and eventually destroy Christian values will work in the case of Islam as well. The problem is one of demographics and the innate persistence of Islam’s propagation and reinforcement system.
This assumes that liberals are actually sincere when they invoke "reason", "science", "human rights", "women's rights", "gay rights", etc. These aren't the Left's end goals, only their tools.
The Left's real agenda is the destruction of traditional western culture and western social institutions. That means fighting religion and traditional social institutions here in America (and in Europe) in the name of the so-called rights and ideals listed above. The same rights and ideals never apply to Third World cultures. "Misogyny" must be opposed if it means that you not willing to pay for a sorority girl's birth control pills, but not such a big issue if Somalis want to give their daughters clitorectomies. Similarly, "Homophobia" must be opposed if some Baptist bakery won't make a cake for a homo-wedding, but not such a problem if Saudis want to decapitate homosexuals or throw them off towers. Along the same lines, liberals never demand that Muslims, African animists, etc abandon their superstitious beliefs in the name of "science," nor will they ever.
What this tells us is that the Left has no real ideals at all other than destroying Western culture. If it's done in the name of "progressive" causes internally, great. If it has to be achieved by an invasion of Third World riffraff, that's great too.