  King, Lu, and Peng recently proved that for $\Delta\geq 4$, any
$K_\Delta$-free graph with maximum degree $\Delta$ has fractional chromatic
number at most $\Delta-\tfrac{2}{67}$ unless it is isomorphic to $C_5\boxtimes
K_2$ or $C_8^2$. Using a different approach we give improved bounds for
$\Delta\geq 6$ and pose several related conjectures. Our proof relies on a
weighted local generalization of the fractional relaxation of Reed's $\omega$,
$\Delta$, $\chi$ conjecture.
