Fractional total colourings of graphs of high girth

Reed conjectured that for every > 0 and ∆ there exists g such that the fractional total chromatic number of a graph with maximum degree ∆ and girth at least g is at most ∆ + 1 + . We prove the conjecture for ∆ = 3 and for even ∆ ≥ 4 in the following stronger form: For each of these values of ∆, there exists g such that the fractional total chromatic number of any graph with maximum degree ∆ and girth at least g is equal to ∆ + 1.