No, because they refuse to acknowledge that the bottom picture happens, and that many of the muslims in the west feel the same (that homosexuals = bad).

You can't judge their logic because they will not accept the fact that muslims, as a whole, don't like them. To draw a comparison, imagine someone tells you they are vehemently against the TPP, yet don't know, and refuse to learn, the provisions (looking at you, Bernie people). To make this analogy more accurate, they simply "know" what some hack of a journalist told them about it. You wouldn't seriously value that person's input becauee they have no idea what they're talking about.

This is like the TPP example. The far left lumped other "minority groups" together such that they can all push common issues together, i.e. the giant "oppressions" that they all face. If you sat one homosexual down and threw all the facts at them about '"moderate Islam" and its view on them, the logical conclusion is that it's incompatible.

I actually had the conversation with a lesbian extended family member who said she wasn't sure how she could reconcile feminism and muslim immigration. She had all the facts, but still wanted someone else to confirn that it was okay to be skeptical of non-westerners who disapprove of western values immigrating here. She's very far left, as are all her friends. She never gave this issue a thought until the Orlando shooting. Things like that wake some people up fast.

Survey on Muslim's view of moral issues: http://www.pewforum.org/2013/04/30/the-worlds-muslims-religion-politics-society-morality/

Poll of British muslims on homosexuality: http://www.telegraph.co.uk/news/2016/04/10/half-of-british-muslims-want-gay-sex-banned-says-poll/