It's problematic because both sides have political backgrounds: The West would like to take advantage of the current chaos to perpetrate a possible revolution. The Iranian government is aware of that, and that regardless of the truth about the death of the girl it's going to be used against it.

The public has reasons not to believe the government story. You have to understand that the "morality police" has nothing to do with morality and it's a result of the power structures making the current Iranian society. In a way, the "morality police" makes the Iranian government weaker and not stronger but it's a ruling tool. This day-to-day ruling tool and its resulting stability comes at the cost of the possible revolution when the general population have had enough.

Will this time be the one? Who knows but at this time the sanctions can come in very handy. (if not for making this successful, they'd be used as leverage of the next negotiations round)