The evidence for this is that the Liberals have lost a fair number of councillors, but only one actual council - the ones where they are already in control are largely being held by them. Presumably the councillors they are losing are in places where a swing to the Conservatives is more likely to unseat Labour control.
The BBC's estimated projected national vote share puts the Tories on 38%, Labour on 23% and the Lib Dems on 28%, which would be a result I'd be very happy with in a national election, as it would push a Lib/Lab coalition, grounded on a move to PR.
