  Local ULIRGs host ubiquitous molecular outflows, including the most massive
and powerful ever detected. These sources have also exceptionally excited
global, galaxy-integrated CO ladders. A connection between outflows and
molecular gas excitation has however never been established, since previous
multi-J CO surveys were limited in spectral resolution and sensitivity and so
could only probe the global molecular gas conditions. We address this question
using new, ground-based, sensitive heterodyne spectroscopy of multiple CO
rotational lines (up to CO(7-6)) in a sample of 17 local ULIRGs. We used the
APEX telescope to survey the CO($J_{up}\geq4$) lines at a high signal-to-noise
ratio, and complemented these data with CO($J_{up}\leq3$) observations
presented in Montoya Arroyave et al. (2023). We detected 74 (out of 75) CO
lines, with up to six transitions per source. Some CO SLEDs peak at
$J_{up}\sim3,4$, which we classify as 'lower excitation', while others plateau
or keep increasing up to the highest-J CO transition probed, and we classify
these as 'higher excitation'. Our analysis includes the results of CO SLED fits
performed with a single large velocity gradient component, but our main focus
is the investigation of possible links between global CO excitation and the
presence of broad and/or high-velocity CO spectral components that can contain
outflowing gas. We discovered an increasing trend of line width as a function
of $J_{up}$ of the CO transition, which is significant at the $4\sigma$ level
and appears to be driven by the eight sources classified as 'higher
excitation'. For such ULIRGs we found that the CO ladders are more excited for
spectral components characterised by higher velocities and/or velocity
dispersion. We favour an interpretation whereby the highly excited CO-emitting
gas in ULIRGs resides in galactic-scale massive molecular outflows.
