If they had their druthers, many if not most liberals including liberal Christians would want to see religious pluralism seize the day. I suppose this in turn is primarily based on the belief that, ultimately, we can't say for sure which religion is true. All religions have some significant measure of truth and validity.
In fact, this may be putting it mildly in some circles. Many think this is the more intellectually astute and even morally enlightened path to trod.
(Of course, there are very vocal militant atheists in the U.S. But their bark is worse than their bite. At least if we judge by the sheer numbers of adherents. I suspect there are probably far more people with liberal beliefs and values who would consider themselves religious or spiritual than there are atheists of any stripe, conservative or liberal or what not.)
However, other societies don't necessarily share this "enlightened" view and attitude. Take the Muslim world. It's not as if many in the Muslim world think all religions are equally valid and true. It's not as if there's a strong liberalizing movement within Islam which seeks to accommodate Islam with other religions like liberal Christianity does. It's not as if many in the Muslim world think we have to best figure out how to modernize Islam and bring it into a pluralistic world. True, there are individual Muslims like Reza Aslan who would doubtless champion such a cause. But my impression is they're few and far between.
As such, when it comes to dialogue between religious liberals like liberal Christians and the majority of Muslims, it seems to me they're talking past one another because they're both entering into dialogue or debate with altogether different assumptions. Spiritual liberals like liberal Christians wish to see Islam become more ecumenical. More open to religious pluralism. More open to a certain amount of relativism with regard to truth. And so forth.
But it seems to me most Muslims are not so subjective in their beliefs and values. It seems to me most Muslims would simply state God exists and has revealed himself. The question for them is which revelation - the Tanakh, the NT, or the Quran? What's more, it seems to me they would likely think this kind of thinking lacks robustness. It's too wishy washy. It isn't serious. It doesn't seek to grapple with objective truth. (Not to mention we're often treated to Muslim clerics and the like denouncing the West for its moral laxity or apathy.)
On the other hand, conservative or Biblical Christianity offers something to Islam which liberals like liberal Christianity doesn't and, indeed, can't.