Commenters "Lord" and Bob Murphy both suggest I look at Ozimek's post on DLR here. It's very good, although I don't know how much it resolves. It goes over a lot of the time-trend issues we've been over here. Wolfer's identification of bias, for example, is similar to what identified as the case that Bob was talking about, which could be a possibility (although one I doubt).
The flows may help to solve these sorts of questions - there are papers on that from Dube and from Meer and West. I'd have to read them. But I can't see how that definitively resolves anything. There can be no rate-of-change change without a change in the flows so the same ambiguities in the changes or lack of changes in the levels is going to be there in the flows too. The difficult task isn't identifying a trend change (which requires a flow change and results in a level change), the difficulty is identifying the right counterfactual trend.
It ultimately boils down to whether we think the time-trends are appropriate or not and if there's an obvious econometric test for it I'm not sure what it is. Time-tends may be wrong, but Occam's razor seems to suggest we should include them (as in DLR). Spatially heterogeneous time trends seem more reasonable than just the right circumstances that would actually introduce bias by including time trends.
Neumark and Wascher suggest we might want non-linear time trends instead of linear ones. One reasonable way to test this is to do an out of sample specification test using the comparison cases. So use a couple specifications of the time trend for periods -12, -11, -10,..., -2, -1, 0, and then figure out which specification best predicts the trend in 1, 2, 3, 4,...,10, 11, 12. Since these cases don't have any dynamic effects of the minimum wage, it should give you a better sense of the non-linearity of time trends. Now, you have to argue that that specification of the time-trend (linear, non-linear, etc.) is also true in the treatment case. But since we're not using the same slopes or parameters itself that seems defensible.
Really I'd need to read Meer and West and the responses but I feel like many of the same points are made here that I made the other day, namely: (1.) time trends should help to reduce bias in most cases, but (2.) you can imagine specific scenarios where the opposite would be the case.
Entirely unsatisfying, eh?
I still think DLR offers the most sensible default - just at first appearances. That doesn't mean there isn't something else going on, but I think it needs to be demonstrated.