  Reed's well-known $\omega$, $\Delta$, $\chi$ conjecture proposes that every
graph satisfies $\chi \leq \lceil \frac 12(\Delta+1+\omega)\rceil$. The second
author formulated a {\em local strengthening} of this conjecture that considers
a bound supplied by the neighbourhood of a single vertex. Following the idea
that the chromatic number cannot be greatly affected by any particular stable
set of vertices, we propose a further strengthening that considers a bound
supplied by the neighbourhoods of two adjacent vertices. We provide some
fundamental evidence in support, namely that the stronger bound holds in the
fractional relaxation and holds for both quasi-line graphs and graphs with
stability number two. We also conjecture that in the fractional version, we can
push the locality even further.
