The evidence probably supports that if the majority of constituents support something that isn't clearly unconstitutional it does tend to happen. See weed legalization in many states which at least in Massachusetts passed in a ballot question with the legislature kicking and screaming through the whole process.
Yet in Mississippi when we had a ballot initiative for medicinal marijuana supported by 2/3rds of everyone, not only did they not pass legislation then (it's tepidly being addressed now, over a year later), but the Mississippi Supreme Court stripped us of our citizen's ballot initiatives because of poorly written legislation in the 90s, which also hasn't been corrected over a year later. But I don't claim to live in a representative democracy these days anyway.
