I don't see why their bases would be against it. It seems to me like a fairly non-ideological belief, that primaries shouldn't be held nearly a year before the election.
Even the state orgs...most of them probably wouldn't be against some kind of reform (other than a few of the early states). While it may be in a particular states' individual interest to have an early primary, it isn't in anyone's interest to have states get in a p1ssing contest with each other and keep moving their primaries up.
The best way to handle this would be a national primary day. The second best way to handle this would be to have maybe three or four possible days on which states can vote and they would pick one.
If the system isn't changed, we'll eventually wind up picking our presidential nominees at the midterm elections.