The last fraction of a fractional conjecture

Reed conjectured that for every $\epsilon >0$ and every integer $\mathrm{\Delta }$, there exists $g$ such that the fractional total chromatic number of every graph with maximum degree $\mathrm{\Delta }$ and girth at least $g$ is at most $\mathrm{\Delta }+1+\epsilon$. The conjecture was proven to be true when $\mathrm{\Delta }=3$ or $\mathrm{\Delta }$ is even. We settle the conjecture by proving it for the remaining cases.