A revival of the girth conjecture

We show that for each ε>0, there exists a number g such that the circular chromatic index of every cubic bridgeless graph with girth at least g is at most 3+ε. This contrasts to the fact (which disproved the Girth conjecture) that there are snarks of arbitrarily large girth. In particular, we show that every cubic bridgeless graph with girth at least 14 has the circular chromatic index at most 7/2.